Dirichlet series
| L(s) = 1 | − 4·3-s − 2·5-s + 4·7-s + 6·9-s + 4·13-s + 8·15-s − 12·17-s − 8·19-s − 16·21-s + 12·23-s + 3·25-s + 4·27-s + 12·29-s − 8·31-s − 8·35-s + 4·37-s − 16·39-s + 12·41-s − 20·43-s − 12·45-s − 12·47-s − 2·49-s + 48·51-s − 12·53-s + 32·57-s + 24·59-s + 4·61-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 0.894·5-s + 1.51·7-s + 2·9-s + 1.10·13-s + 2.06·15-s − 2.91·17-s − 1.83·19-s − 3.49·21-s + 2.50·23-s + 3/5·25-s + 0.769·27-s + 2.22·29-s − 1.43·31-s − 1.35·35-s + 0.657·37-s − 2.56·39-s + 1.87·41-s − 3.04·43-s − 1.78·45-s − 1.75·47-s − 2/7·49-s + 6.72·51-s − 1.64·53-s + 4.23·57-s + 3.12·59-s + 0.512·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(400\) = \(2^{4} \cdot 5^{2}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0255043\) |
| Root analytic conductor: | \(0.399625\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 400,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2215859706\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2215859706\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | ||
| 5 | $C_1$ | \( ( 1 + T )^{2} \) | ||
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.3.e_k |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.7.ae_s | |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.11.a_w | |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.13.ae_be | |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.17.m_cs | |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.19.i_cc | |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | 2.23.am_de | |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | 2.29.am_dq | |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | 2.31.i_da | |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.37.ae_da | |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | 2.41.am_eo | |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) | 2.43.u_he | |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.47.m_fa | |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.53.m_fm | |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) | 2.59.ay_kc | |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.61.ae_ew | |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.67.ae_fi | |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) | 2.71.y_la | |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.73.ae_fu | |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) | 2.79.aq_io | |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | 2.83.am_hu | |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | 2.89.m_ig | |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | 2.97.ae_hq | |
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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
−19.6035760866, −19.6035760866, −18.1271877078, −18.1271877078, −17.3081878781, −17.3081878781, −16.0995603957, −16.0995603957, −14.7919331095, −14.7919331095, −13.0023664158, −13.0023664158, −11.5085445320, −11.5085445320, −10.8347395030, −10.8347395030, −8.55221755078, −8.55221755078, −6.57891116466, −6.57891116466, −4.78130792718, −4.78130792718, 4.78130792718, 4.78130792718, 6.57891116466, 6.57891116466, 8.55221755078, 8.55221755078, 10.8347395030, 10.8347395030, 11.5085445320, 11.5085445320, 13.0023664158, 13.0023664158, 14.7919331095, 14.7919331095, 16.0995603957, 16.0995603957, 17.3081878781, 17.3081878781, 18.1271877078, 18.1271877078, 19.6035760866, 19.6035760866