Dirichlet series
| L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s − 2·5-s − 4·6-s − 2·9-s + 4·10-s − 2·11-s + 4·12-s − 4·15-s − 4·16-s + 2·17-s + 4·18-s + 6·19-s − 4·20-s + 4·22-s + 10·23-s − 3·25-s − 10·27-s − 10·29-s + 8·30-s + 6·31-s + 8·32-s − 4·33-s − 4·34-s − 4·36-s − 6·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 4-s − 0.894·5-s − 1.63·6-s − 2/3·9-s + 1.26·10-s − 0.603·11-s + 1.15·12-s − 1.03·15-s − 16-s + 0.485·17-s + 0.942·18-s + 1.37·19-s − 0.894·20-s + 0.852·22-s + 2.08·23-s − 3/5·25-s − 1.92·27-s − 1.85·29-s + 1.46·30-s + 1.07·31-s + 1.41·32-s − 0.696·33-s − 0.685·34-s − 2/3·36-s − 0.986·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(132496\) = \(2^{4} \cdot 7^{2} \cdot 13^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(8.44805\) |
| Root analytic conductor: | \(1.70486\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 132496,\ (\ :1/2, 1/2),\ -1)\) |
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 | 2 | $C_2$ | \( 1 + p T + p T^{2} \) | |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | ||
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) | 2.3.ac_g |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.5.c_h | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.11.c_ac | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.17.ac_ba | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) | 2.19.ag_br | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) | 2.23.ak_cp | |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) | 2.29.k_df | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.31.ag_bj | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.37.g_de | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.41.e_cs | |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) | 2.43.c_dj | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.47.c_bf | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.53.g_db | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) | 2.59.ai_eo | |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.61.ae_as | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.67.q_hm | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) | 2.71.ag_be | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) | 2.73.k_ed | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.79.c_fn | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.83.ak_dn | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.89.g_fv | |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.97.ag_hf | |
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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
−13.8601618885, −13.5240709135, −13.5080204842, −12.6926896247, −12.1334996147, −11.5927992260, −11.2040338670, −11.1518658275, −10.2510418534, −9.97759541606, −9.38463950725, −8.99834677241, −8.73547122738, −8.14438866848, −7.88305646626, −7.43563929744, −7.14918000367, −6.37684020437, −5.34298608831, −5.27883313771, −4.23622923362, −3.39334742454, −3.11744276397, −2.38361565588, −1.35836937710, 0, 1.35836937710, 2.38361565588, 3.11744276397, 3.39334742454, 4.23622923362, 5.27883313771, 5.34298608831, 6.37684020437, 7.14918000367, 7.43563929744, 7.88305646626, 8.14438866848, 8.73547122738, 8.99834677241, 9.38463950725, 9.97759541606, 10.2510418534, 11.1518658275, 11.2040338670, 11.5927992260, 12.1334996147, 12.6926896247, 13.5080204842, 13.5240709135, 13.8601618885