Dirichlet series
| L(s) = 1 | − 2·4-s − 2·5-s + 2·7-s − 2·9-s + 4·11-s + 2·13-s + 4·16-s − 8·17-s − 6·19-s + 4·20-s + 10·23-s − 3·25-s − 4·28-s − 14·29-s + 14·31-s − 4·35-s + 4·36-s − 4·41-s − 2·43-s − 8·44-s + 4·45-s − 6·47-s + 3·49-s − 4·52-s − 6·53-s − 8·55-s + 4·61-s + ⋯ |
| L(s) = 1 | − 4-s − 0.894·5-s + 0.755·7-s − 2/3·9-s + 1.20·11-s + 0.554·13-s + 16-s − 1.94·17-s − 1.37·19-s + 0.894·20-s + 2.08·23-s − 3/5·25-s − 0.755·28-s − 2.59·29-s + 2.51·31-s − 0.676·35-s + 2/3·36-s − 0.624·41-s − 0.304·43-s − 1.20·44-s + 0.596·45-s − 0.875·47-s + 3/7·49-s − 0.554·52-s − 0.824·53-s − 1.07·55-s + 0.512·61-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 | $C_2$ | \( 1 + p T^{2} \) | |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) | ||
| 13 | $C_1$ | \( ( 1 - T )^{2} \) | ||
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.3.a_c |
| 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$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) | 2.11.ae_w | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.17.i_bu | |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.19.g_bf | |
| 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$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.29.o_dz | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) | 2.31.ao_ed | |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.37.a_cs | |
| 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$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.47.g_cp | |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.53.g_db | |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | 2.59.a_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$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.67.ae_ag | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.71.ai_cg | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) | 2.73.ao_gx | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.79.g_gh | |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.83.c_fv | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.89.ag_br | |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) | 2.97.c_hn | |
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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, −13.3285947702, −12.6926896247, −12.4014804350, −11.5927992260, −11.3434326840, −11.2040338670, −10.7326741027, −9.97759541606, −9.38463950725, −8.96547406443, −8.73547122738, −8.14438866848, −8.02303408131, −7.14918000367, −6.48432181419, −6.37684020437, −5.34298608831, −4.95454046108, −4.23622923362, −4.04609643315, −3.39334742454, −2.38361565588, −1.35836937710, 0, 1.35836937710, 2.38361565588, 3.39334742454, 4.04609643315, 4.23622923362, 4.95454046108, 5.34298608831, 6.37684020437, 6.48432181419, 7.14918000367, 8.02303408131, 8.14438866848, 8.73547122738, 8.96547406443, 9.38463950725, 9.97759541606, 10.7326741027, 11.2040338670, 11.3434326840, 11.5927992260, 12.4014804350, 12.6926896247, 13.3285947702, 13.5240709135, 13.8601618885