Dirichlet series
| L(s) = 1 | − 3·3-s + 4-s + 2·5-s + 2·7-s + 4·9-s − 4·11-s − 3·12-s − 6·13-s − 6·15-s + 16-s − 4·17-s − 2·19-s + 2·20-s − 6·21-s − 14·23-s − 3·25-s + 2·28-s + 2·29-s + 4·31-s + 12·33-s + 4·35-s + 4·36-s − 12·37-s + 18·39-s + 2·41-s − 2·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1/2·4-s + 0.894·5-s + 0.755·7-s + 4/3·9-s − 1.20·11-s − 0.866·12-s − 1.66·13-s − 1.54·15-s + 1/4·16-s − 0.970·17-s − 0.458·19-s + 0.447·20-s − 1.30·21-s − 2.91·23-s − 3/5·25-s + 0.377·28-s + 0.371·29-s + 0.718·31-s + 2.08·33-s + 0.676·35-s + 2/3·36-s − 1.97·37-s + 2.88·39-s + 0.312·41-s − 0.304·43-s − 0.603·44-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 33708 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33708 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(33708\) = \(2^{2} \cdot 3 \cdot 53^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(2.14925\) |
| Root analytic conductor: | \(1.21079\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 33708,\ (\ :1/2, 1/2),\ -1)\) |
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) | |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) | ||
| 53 | $C_1$ | \( ( 1 + T )^{2} \) | ||
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.5.ac_h |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) | 2.7.ac_o | |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) | 2.11.e_z | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.13.g_bi | |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) | 2.17.e_n | |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.19.c_be | |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) | 2.23.o_dn | |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.29.ac_bi | |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.31.ae_cf | |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.37.m_dq | |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.41.ac_cg | |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) | 2.43.c_dj | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) | 2.47.ag_dq | |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) | 2.59.ay_jt | |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) | 2.61.a_w | |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) | 2.67.g_fm | |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) | 2.71.am_fm | |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) | 2.73.as_is | |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) | 2.79.am_gn | |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.83.a_fa | |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \) | 2.89.ai_gn | |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) | 2.97.ba_nz | |
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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
−15.7293418305, −15.0050722611, −14.4426941234, −13.8387373986, −13.7537623204, −12.7718926136, −12.5526910886, −11.8876937856, −11.7905075056, −11.2201515442, −10.5783128085, −10.2561820708, −10.0193514239, −9.39656287095, −8.29725987978, −8.06909254484, −7.37901993255, −6.48615429970, −6.46531318491, −5.55911961841, −5.24864413998, −4.80952345157, −3.94088749994, −2.25107714074, −2.15260824388, 0, 2.15260824388, 2.25107714074, 3.94088749994, 4.80952345157, 5.24864413998, 5.55911961841, 6.46531318491, 6.48615429970, 7.37901993255, 8.06909254484, 8.29725987978, 9.39656287095, 10.0193514239, 10.2561820708, 10.5783128085, 11.2201515442, 11.7905075056, 11.8876937856, 12.5526910886, 12.7718926136, 13.7537623204, 13.8387373986, 14.4426941234, 15.0050722611, 15.7293418305