Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 7-s − 3·8-s − 11-s − 2·12-s − 2·13-s + 14-s + 16-s − 4·17-s − 2·19-s + 2·21-s + 22-s + 4·23-s + 6·24-s − 25-s + 2·26-s + 2·27-s − 28-s + 2·31-s + 32-s + 2·33-s + 4·34-s − 9·37-s + 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s − 1.06·8-s − 0.301·11-s − 0.577·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.458·19-s + 0.436·21-s + 0.213·22-s + 0.834·23-s + 1.22·24-s − 1/5·25-s + 0.392·26-s + 0.384·27-s − 0.188·28-s + 0.359·31-s + 0.176·32-s + 0.348·33-s + 0.685·34-s − 1.47·37-s + 0.324·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 7675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(7675\) = \(5^{2} \cdot 307\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.489364\) |
| Root analytic conductor: | \(0.836388\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 7675,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_2$ | \( 1 + T^{2} \) | |
| 307 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 28 T + p T^{2} ) \) | ||
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) | 2.2.b_a |
| 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.3.c_e | |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.7.b_k | |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.11.b_ak | |
| 13 | $D_{4}$ | \( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.13.c_ak | |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.17.e_bm | |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.19.c_w | |
| 23 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.23.ae_w | |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) | 2.29.a_ac | |
| 31 | $D_{4}$ | \( 1 - 2 T - 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.31.ac_am | |
| 37 | $D_{4}$ | \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.37.j_ce | |
| 41 | $D_{4}$ | \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.41.af_bc | |
| 43 | $D_{4}$ | \( 1 + 16 T + 132 T^{2} + 16 p T^{3} + p^{2} T^{4} \) | 2.43.q_fc | |
| 47 | $D_{4}$ | \( 1 - 8 T + 64 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.47.ai_cm | |
| 53 | $D_{4}$ | \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.53.ad_dw | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.59.e_w | |
| 61 | $D_{4}$ | \( 1 - 2 T - 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.61.ac_acs | |
| 67 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.67.c_cs | |
| 71 | $D_{4}$ | \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) | 2.71.ak_cc | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.73.ak_gg | |
| 79 | $D_{4}$ | \( 1 + 5 T + 50 T^{2} + 5 p T^{3} + p^{2} T^{4} \) | 2.79.f_by | |
| 83 | $D_{4}$ | \( 1 + 11 T + 114 T^{2} + 11 p T^{3} + p^{2} T^{4} \) | 2.83.l_ek | |
| 89 | $D_{4}$ | \( 1 - 7 T + 68 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.89.ah_cq | |
| 97 | $D_{4}$ | \( 1 - 17 T + 220 T^{2} - 17 p T^{3} + p^{2} T^{4} \) | 2.97.ar_im | |
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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
−17.2270858345, −16.8418166040, −16.3758056815, −15.7110630233, −15.3327939952, −15.0167953310, −14.1530125485, −13.6213593921, −12.8449372364, −12.5445363135, −11.7982142124, −11.4533799911, −11.0837668238, −10.3346235318, −9.95884184593, −9.10148433403, −8.79216057796, −8.06457593620, −7.14049806195, −6.59027098548, −6.16852589240, −5.32695939851, −4.72935663011, −3.38855539194, −2.32820685683, 0, 2.32820685683, 3.38855539194, 4.72935663011, 5.32695939851, 6.16852589240, 6.59027098548, 7.14049806195, 8.06457593620, 8.79216057796, 9.10148433403, 9.95884184593, 10.3346235318, 11.0837668238, 11.4533799911, 11.7982142124, 12.5445363135, 12.8449372364, 13.6213593921, 14.1530125485, 15.0167953310, 15.3327939952, 15.7110630233, 16.3758056815, 16.8418166040, 17.2270858345