Dirichlet series
| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 6·11-s − 4·13-s + 2·14-s + 16-s + 6·17-s + 3·19-s − 6·22-s + 6·23-s + 2·25-s − 4·26-s + 2·28-s + 2·31-s + 32-s + 6·34-s + 2·37-s + 3·38-s − 3·41-s + 9·43-s − 6·44-s + 6·46-s − 12·47-s − 49-s + 2·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 1.80·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.688·19-s − 1.27·22-s + 1.25·23-s + 2/5·25-s − 0.784·26-s + 0.377·28-s + 0.359·31-s + 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.486·38-s − 0.468·41-s + 1.37·43-s − 0.904·44-s + 0.884·46-s − 1.75·47-s − 1/7·49-s + 0.282·50-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(373248\) = \(2^{9} \cdot 3^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(23.7986\) |
| Root analytic conductor: | \(2.20870\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 373248,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2.938985184\) |
| \(L(\frac12)\) | \(\approx\) | \(2.938985184\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ | |
|---|---|---|---|---|
| bad | 2 | $C_1$ | \( 1 - T \) | |
| 3 | \( 1 \) | |||
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) | 2.5.a_ac |
| 7 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.7.ac_f | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.11.g_w | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.13.e_o | |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.17.ag_t | |
| 19 | $D_{4}$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.19.ad_ai | |
| 23 | $D_{4}$ | \( 1 - 6 T + 49 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.23.ag_bx | |
| 29 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) | 2.29.a_bc | |
| 31 | $D_{4}$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.31.ac_ah | |
| 37 | $D_{4}$ | \( 1 - 2 T - 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.37.ac_ak | |
| 41 | $D_{4}$ | \( 1 + 3 T + 76 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.41.d_cy | |
| 43 | $D_{4}$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.43.aj_bo | |
| 47 | $D_{4}$ | \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) | 2.47.m_ef | |
| 53 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.53.g_bc | |
| 59 | $D_{4}$ | \( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} \) | 2.59.ap_fs | |
| 61 | $D_{4}$ | \( 1 + 4 T + 44 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.61.e_bs | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.67.ae_dy | |
| 71 | $D_{4}$ | \( 1 + 6 T + T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.71.g_b | |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.73.d_cy | |
| 79 | $D_{4}$ | \( 1 - 14 T + 101 T^{2} - 14 p T^{3} + p^{2} T^{4} \) | 2.79.ao_dx | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.83.j_gk | |
| 89 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) | 2.89.am_fa | |
| 97 | $D_{4}$ | \( 1 - 18 T + 166 T^{2} - 18 p T^{3} + p^{2} T^{4} \) | 2.97.as_gk | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.8952882724, −12.6131253668, −12.0778483773, −11.7262647951, −11.3367721949, −10.8668580644, −10.4638078072, −10.0762462388, −9.71008861684, −9.21301436352, −8.48531070042, −8.09582399785, −7.61439928110, −7.47191799645, −6.94180385153, −6.24574964428, −5.64008978985, −5.22053064604, −4.86849807357, −4.66566460267, −3.60555159036, −3.07672039174, −2.65492617114, −1.92661784880, −0.876947519158, 0.876947519158, 1.92661784880, 2.65492617114, 3.07672039174, 3.60555159036, 4.66566460267, 4.86849807357, 5.22053064604, 5.64008978985, 6.24574964428, 6.94180385153, 7.47191799645, 7.61439928110, 8.09582399785, 8.48531070042, 9.21301436352, 9.71008861684, 10.0762462388, 10.4638078072, 10.8668580644, 11.3367721949, 11.7262647951, 12.0778483773, 12.6131253668, 12.8952882724