Dirichlet series
| L(s) = 1 | − 2·4-s + 9-s − 5·11-s + 3·19-s + 8·29-s − 2·31-s − 2·36-s − 2·41-s + 10·44-s + 8·49-s − 11·59-s − 5·61-s + 8·64-s − 8·71-s − 6·76-s + 6·79-s − 8·81-s − 5·89-s − 5·99-s + 101-s + 103-s + 107-s + 109-s + 113-s − 16·116-s − 2·121-s + 4·124-s + ⋯ |
| L(s) = 1 | − 4-s + 1/3·9-s − 1.50·11-s + 0.688·19-s + 1.48·29-s − 0.359·31-s − 1/3·36-s − 0.312·41-s + 1.50·44-s + 8/7·49-s − 1.43·59-s − 0.640·61-s + 64-s − 0.949·71-s − 0.688·76-s + 0.675·79-s − 8/9·81-s − 0.529·89-s − 0.502·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.48·116-s − 0.181·121-s + 0.359·124-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(180625\) = \(5^{4} \cdot 17^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(11.5168\) |
| Root analytic conductor: | \(1.84218\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 180625,\ (\ :1/2, 1/2),\ -1)\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | 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 | 5 | \( 1 \) | ||
| 17 | $C_2$ | \( 1 + T^{2} \) | ||
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) | 2.2.a_c |
| 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) | 2.3.a_ab | |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) | 2.7.a_ai | |
| 11 | $D_{4}$ | \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.11.f_bb | |
| 13 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) | 2.13.a_n | |
| 19 | $D_{4}$ | \( 1 - 3 T + 39 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.19.ad_bn | |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.23.a_k | |
| 29 | $D_{4}$ | \( 1 - 8 T + 69 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.29.ai_cr | |
| 31 | $D_{4}$ | \( 1 + 2 T + 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.31.c_br | |
| 37 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) | 2.37.a_abo | |
| 41 | $D_{4}$ | \( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.41.c_bm | |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) | 2.43.a_bj | |
| 47 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) | 2.47.a_acc | |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.53.a_cs | |
| 59 | $D_{4}$ | \( 1 + 11 T + 137 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.59.l_fh | |
| 61 | $D_{4}$ | \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.61.f_bb | |
| 67 | $C_2^2$ | \( 1 - 63 T^{2} + p^{2} T^{4} \) | 2.67.a_acl | |
| 71 | $D_{4}$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.71.i_bh | |
| 73 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) | 2.73.a_aeb | |
| 79 | $D_{4}$ | \( 1 - 6 T + 87 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.79.ag_dj | |
| 83 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) | 2.83.a_acs | |
| 89 | $D_{4}$ | \( 1 + 5 T + 83 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.89.f_df | |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.97.a_by | |
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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
−13.5820469222, −13.4505052359, −12.9337995376, −12.5527918208, −12.0628137669, −11.7556468570, −11.0048250033, −10.6213729092, −10.3216213876, −9.81143513061, −9.35379957053, −8.99296505629, −8.37200325299, −8.08749726341, −7.54679603878, −7.08751938401, −6.50402999413, −5.79384107264, −5.33274685649, −4.84299263197, −4.43504279524, −3.78527185574, −2.96995958514, −2.47813955610, −1.28397670944, 0, 1.28397670944, 2.47813955610, 2.96995958514, 3.78527185574, 4.43504279524, 4.84299263197, 5.33274685649, 5.79384107264, 6.50402999413, 7.08751938401, 7.54679603878, 8.08749726341, 8.37200325299, 8.99296505629, 9.35379957053, 9.81143513061, 10.3216213876, 10.6213729092, 11.0048250033, 11.7556468570, 12.0628137669, 12.5527918208, 12.9337995376, 13.4505052359, 13.5820469222