Dirichlet series
| L(s) = 1 | − 4.19e6·2-s − 1.29e10·3-s + 1.31e13·4-s − 3.98e14·5-s + 5.44e16·6-s + 1.17e18·7-s − 3.68e19·8-s − 4.86e20·9-s + 1.67e21·10-s − 1.16e22·11-s − 1.71e23·12-s − 1.65e24·13-s − 4.92e24·14-s + 5.17e24·15-s + 9.67e25·16-s + 3.43e26·17-s + 2.04e27·18-s − 5.26e26·19-s − 5.26e27·20-s − 1.52e28·21-s + 4.87e28·22-s + 2.29e29·23-s + 4.78e29·24-s + 1.34e30·25-s + 6.95e30·26-s + 1.05e31·27-s + 1.55e31·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.716·3-s + 3/2·4-s − 0.373·5-s + 1.01·6-s + 0.795·7-s − 1.41·8-s − 1.48·9-s + 0.528·10-s − 0.473·11-s − 1.07·12-s − 1.86·13-s − 1.12·14-s + 0.267·15-s + 5/4·16-s + 1.20·17-s + 2.09·18-s − 0.169·19-s − 0.560·20-s − 0.569·21-s + 0.669·22-s + 1.21·23-s + 1.01·24-s + 1.18·25-s + 2.63·26-s + 1.77·27-s + 1.19·28-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr
=\mathstrut & \, \Lambda(44-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+43/2)^{2} \, L(s)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(4\) |
| \( N \) | = | \(4\) = \(2^{2}\) |
| \( \varepsilon \) | = | $1$ |
| motivic weight | = | \(43\) |
| character | : | induced by $\chi_{2} (1, \cdot )$ |
| primitive | : | no |
| self-dual | : | yes |
| analytic rank | = | 0 |
| Selberg data | = | $(4,\ 4,\ (\ :43/2, 43/2),\ 1)$ |
| $L(22)$ | $\approx$ | $0.840322$ |
| $L(\frac12)$ | $\approx$ | $0.840322$ |
| $L(\frac{45}{2})$ | not available | |
| $L(1)$ | not available |
Euler product
\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]
where, for $p \neq 2$,
\(F_p\) is a polynomial of degree 4. If $p = 2$, then $F_p$ is a polynomial of degree at most 3.
| $p$ | $\Gal(F_p)$ | $F_p$ | |
|---|---|---|---|
| bad | 2 | $C_1$ | \( ( 1 + p^{21} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 4327210328 p T + 299551664144015314 p^{7} T^{2} + 4327210328 p^{44} T^{3} + p^{86} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 637860338052 p^{4} T - \)\(24\!\cdots\!58\)\( p^{11} T^{2} + 637860338052 p^{47} T^{3} + p^{86} T^{4} \) | |
| 7 | $D_{4}$ | \( 1 - 1174870033543241008 T + \)\(91\!\cdots\!14\)\( p^{3} T^{2} - 1174870033543241008 p^{43} T^{3} + p^{86} T^{4} \) | |
| 11 | $D_{4}$ | \( 1 + \)\(10\!\cdots\!16\)\( p T + \)\(32\!\cdots\!86\)\( p^{2} T^{2} + \)\(10\!\cdots\!16\)\( p^{44} T^{3} + p^{86} T^{4} \) | |
| 13 | $D_{4}$ | \( 1 + \)\(12\!\cdots\!68\)\( p T + \)\(74\!\cdots\!78\)\( p^{4} T^{2} + \)\(12\!\cdots\!68\)\( p^{44} T^{3} + p^{86} T^{4} \) | |
| 17 | $D_{4}$ | \( 1 - \)\(20\!\cdots\!04\)\( p T + \)\(32\!\cdots\!14\)\( p^{3} T^{2} - \)\(20\!\cdots\!04\)\( p^{44} T^{3} + p^{86} T^{4} \) | |
| 19 | $D_{4}$ | \( 1 + \)\(52\!\cdots\!00\)\( T + \)\(65\!\cdots\!22\)\( p T^{2} + \)\(52\!\cdots\!00\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 23 | $D_{4}$ | \( 1 - \)\(22\!\cdots\!36\)\( T + \)\(31\!\cdots\!46\)\( p T^{2} - \)\(22\!\cdots\!36\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 29 | $D_{4}$ | \( 1 - \)\(11\!\cdots\!80\)\( p T + \)\(19\!\cdots\!58\)\( p^{2} T^{2} - \)\(11\!\cdots\!80\)\( p^{44} T^{3} + p^{86} T^{4} \) | |
| 31 | $D_{4}$ | \( 1 - \)\(16\!\cdots\!04\)\( T + \)\(10\!\cdots\!06\)\( p T^{2} - \)\(16\!\cdots\!04\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 37 | $D_{4}$ | \( 1 + \)\(11\!\cdots\!36\)\( p T + \)\(21\!\cdots\!98\)\( p^{2} T^{2} + \)\(11\!\cdots\!36\)\( p^{44} T^{3} + p^{86} T^{4} \) | |
| 41 | $D_{4}$ | \( 1 - \)\(17\!\cdots\!64\)\( p T + \)\(34\!\cdots\!06\)\( p^{2} T^{2} - \)\(17\!\cdots\!64\)\( p^{44} T^{3} + p^{86} T^{4} \) | |
| 43 | $D_{4}$ | \( 1 - \)\(23\!\cdots\!36\)\( T + \)\(41\!\cdots\!38\)\( T^{2} - \)\(23\!\cdots\!36\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 47 | $D_{4}$ | \( 1 + \)\(39\!\cdots\!92\)\( T + \)\(27\!\cdots\!62\)\( T^{2} + \)\(39\!\cdots\!92\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 53 | $D_{4}$ | \( 1 - \)\(21\!\cdots\!56\)\( T + \)\(24\!\cdots\!38\)\( T^{2} - \)\(21\!\cdots\!56\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 59 | $D_{4}$ | \( 1 - \)\(67\!\cdots\!40\)\( T + \)\(20\!\cdots\!58\)\( T^{2} - \)\(67\!\cdots\!40\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 61 | $D_{4}$ | \( 1 - \)\(10\!\cdots\!04\)\( T + \)\(94\!\cdots\!66\)\( T^{2} - \)\(10\!\cdots\!04\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 67 | $D_{4}$ | \( 1 + \)\(12\!\cdots\!52\)\( T + \)\(36\!\cdots\!02\)\( T^{2} + \)\(12\!\cdots\!52\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 71 | $D_{4}$ | \( 1 - \)\(10\!\cdots\!64\)\( T + \)\(10\!\cdots\!46\)\( T^{2} - \)\(10\!\cdots\!64\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 73 | $D_{4}$ | \( 1 + \)\(10\!\cdots\!24\)\( T + \)\(26\!\cdots\!78\)\( T^{2} + \)\(10\!\cdots\!24\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 79 | $D_{4}$ | \( 1 + \)\(66\!\cdots\!20\)\( T + \)\(90\!\cdots\!78\)\( T^{2} + \)\(66\!\cdots\!20\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 83 | $D_{4}$ | \( 1 - \)\(54\!\cdots\!56\)\( T + \)\(14\!\cdots\!58\)\( T^{2} - \)\(54\!\cdots\!56\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 89 | $D_{4}$ | \( 1 - \)\(29\!\cdots\!20\)\( T + \)\(85\!\cdots\!38\)\( T^{2} - \)\(29\!\cdots\!20\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
| 97 | $D_{4}$ | \( 1 + \)\(22\!\cdots\!12\)\( T + \)\(34\!\cdots\!82\)\( T^{2} + \)\(22\!\cdots\!12\)\( p^{43} T^{3} + p^{86} T^{4} \) | |
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\[\begin{aligned}
L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]