Dirichlet series
| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s − 3·9-s − 5·11-s − 12-s − 3·13-s + 2·14-s + 16-s − 17-s + 3·18-s + 2·21-s + 5·22-s + 2·23-s + 24-s − 4·25-s + 3·26-s + 4·27-s − 2·28-s − 3·29-s − 31-s − 32-s + 5·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s − 9-s − 1.50·11-s − 0.288·12-s − 0.832·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.707·18-s + 0.436·21-s + 1.06·22-s + 0.417·23-s + 0.204·24-s − 4/5·25-s + 0.588·26-s + 0.769·27-s − 0.377·28-s − 0.557·29-s − 0.179·31-s − 0.176·32-s + 0.870·33-s + 0.171·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 9736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(9736\) = \(2^{3} \cdot 1217\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.620775\) |
| Root analytic conductor: | \(0.887633\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 9736,\ (\ :1/2, 1/2),\ -1)\) |
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 | $C_1$ | \( 1 + T \) | |
| 1217 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 9 T + p T^{2} ) \) | ||
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) | 2.3.b_e |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) | 2.5.a_e | |
| 7 | $D_{4}$ | \( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} \) | 2.7.c_a | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) | 2.11.f_w | |
| 13 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.13.d_k | |
| 17 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.17.b_k | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.23.ac_k | |
| 29 | $D_{4}$ | \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.29.d_br | |
| 31 | $D_{4}$ | \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.31.b_o | |
| 37 | $D_{4}$ | \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.37.ad_ca | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) | 2.41.at_gk | |
| 43 | $D_{4}$ | \( 1 + 9 T + 67 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.43.j_cp | |
| 47 | $D_{4}$ | \( 1 - 5 T - 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.47.af_ar | |
| 53 | $D_{4}$ | \( 1 - 2 T - 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.53.ac_abg | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.59.d_cm | |
| 61 | $D_{4}$ | \( 1 + 17 T + 165 T^{2} + 17 p T^{3} + p^{2} T^{4} \) | 2.61.r_gj | |
| 67 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.67.g_bo | |
| 71 | $D_{4}$ | \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \) | 2.71.i_ec | |
| 73 | $D_{4}$ | \( 1 - 7 T + 87 T^{2} - 7 p T^{3} + p^{2} T^{4} \) | 2.73.ah_dj | |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) | 2.79.j_ec | |
| 83 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.83.d_bo | |
| 89 | $D_{4}$ | \( 1 - 4 T + 136 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.89.ae_fg | |
| 97 | $D_{4}$ | \( 1 - 5 T + 20 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.97.af_u | |
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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
−16.8588999301, −16.4891723029, −16.0603089801, −15.4455029775, −15.1439823392, −14.4730433105, −13.8940967330, −13.2332152066, −12.8076281896, −12.2959240491, −11.6693135480, −11.2246410715, −10.5852189715, −10.3402566433, −9.44688823357, −9.21462225364, −8.41303852606, −7.57289365625, −7.49581796552, −6.40419471808, −5.82950719943, −5.38035700086, −4.39969735327, −3.05822595856, −2.45360253559, 0, 2.45360253559, 3.05822595856, 4.39969735327, 5.38035700086, 5.82950719943, 6.40419471808, 7.49581796552, 7.57289365625, 8.41303852606, 9.21462225364, 9.44688823357, 10.3402566433, 10.5852189715, 11.2246410715, 11.6693135480, 12.2959240491, 12.8076281896, 13.2332152066, 13.8940967330, 14.4730433105, 15.1439823392, 15.4455029775, 16.0603089801, 16.4891723029, 16.8588999301