Dirichlet series
| L(s) = 1 | − 2-s − 2·3-s + 4-s − 4·5-s + 2·6-s + 7-s + 8-s + 2·9-s + 4·10-s + 2·11-s − 2·12-s − 2·13-s − 14-s + 8·15-s − 3·16-s − 2·17-s − 2·18-s + 19-s − 4·20-s − 2·21-s − 2·22-s + 5·23-s − 2·24-s + 6·25-s + 2·26-s − 6·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 2/3·9-s + 1.26·10-s + 0.603·11-s − 0.577·12-s − 0.554·13-s − 0.267·14-s + 2.06·15-s − 3/4·16-s − 0.485·17-s − 0.471·18-s + 0.229·19-s − 0.894·20-s − 0.436·21-s − 0.426·22-s + 1.04·23-s − 0.408·24-s + 6/5·25-s + 0.392·26-s − 1.15·27-s + 0.188·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 394 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(394\) = \(2 \cdot 197\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0251217\) |
| Root analytic conductor: | \(0.398118\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((4,\ 394,\ (\ :1/2, 1/2),\ 1)\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.2007827436\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2007827436\) |
| \(L(\frac{3}{2})\) | 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$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) | |
| 197 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 18 T + p T^{2} ) \) | ||
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.3.c_c |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.5.e_k | |
| 7 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.7.ab_ac | |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.11.ac_c | |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.13.c_w | |
| 17 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.17.c_ag | |
| 19 | $D_{4}$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.19.ab_aw | |
| 23 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.23.af_be | |
| 29 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | 2.29.d_bc | |
| 31 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | 2.31.ag_be | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.37.h_bs | |
| 41 | $C_2^2$ | \( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.41.ad_bs | |
| 43 | $D_{4}$ | \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.43.ab_ag | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.47.h_dq | |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.53.a_cs | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.59.am_eo | |
| 61 | $D_{4}$ | \( 1 + T + 16 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.61.b_q | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) | 2.67.c_o | |
| 71 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.71.ac_dy | |
| 73 | $D_{4}$ | \( 1 - 2 T - 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) | 2.73.ac_ack | |
| 79 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) | 2.79.a_abq | |
| 83 | $D_{4}$ | \( 1 - 9 T + 66 T^{2} - 9 p T^{3} + p^{2} T^{4} \) | 2.83.aj_co | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.89.i_gc | |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | 2.97.e_hq | |
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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
−19.6295715505, −19.3059469824, −19.0248356528, −17.9947296559, −17.5176256677, −16.8487865706, −16.5231674940, −15.6004576954, −15.4443043074, −14.5647329690, −13.5078710086, −12.5489303651, −11.7483015383, −11.4806467239, −10.9979364628, −10.0972223242, −9.01903323173, −8.03781843767, −7.36037006983, −6.65135433456, −5.09410049950, −4.06198103368, 4.06198103368, 5.09410049950, 6.65135433456, 7.36037006983, 8.03781843767, 9.01903323173, 10.0972223242, 10.9979364628, 11.4806467239, 11.7483015383, 12.5489303651, 13.5078710086, 14.5647329690, 15.4443043074, 15.6004576954, 16.5231674940, 16.8487865706, 17.5176256677, 17.9947296559, 19.0248356528, 19.3059469824, 19.6295715505