Dirichlet series
| L(s) = 1 | − 4-s − 3·5-s − 2·7-s − 2·8-s − 2·9-s + 4·11-s − 6·13-s + 16-s + 3·20-s − 23-s + 2·25-s − 3·27-s + 2·28-s − 9·29-s + 5·31-s + 4·32-s + 6·35-s + 2·36-s − 4·37-s + 6·40-s + 41-s + 43-s − 4·44-s + 6·45-s + 10·47-s + 2·49-s + 6·52-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.34·5-s − 0.755·7-s − 0.707·8-s − 2/3·9-s + 1.20·11-s − 1.66·13-s + 1/4·16-s + 0.670·20-s − 0.208·23-s + 2/5·25-s − 0.577·27-s + 0.377·28-s − 1.67·29-s + 0.898·31-s + 0.707·32-s + 1.01·35-s + 1/3·36-s − 0.657·37-s + 0.948·40-s + 0.156·41-s + 0.152·43-s − 0.603·44-s + 0.894·45-s + 1.45·47-s + 2/7·49-s + 0.832·52-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 10110 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10110 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(4\) |
| Conductor: | \(10110\) = \(2 \cdot 3 \cdot 5 \cdot 337\) |
| Sign: | $-1$ |
| Analytic conductor: | \(0.644622\) |
| Root analytic conductor: | \(0.896037\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((4,\ 10110,\ (\ :1/2, 1/2),\ -1)\) |
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_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) | |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) | ||
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) | ||
| 337 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 14 T + p T^{2} ) \) | ||
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | 2.7.c_c |
| 11 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.11.ae_o | |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) | 2.13.g_ba | |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) | 2.17.a_c | |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | 2.19.a_k | |
| 23 | $D_{4}$ | \( 1 + T - 18 T^{2} + p T^{3} + p^{2} T^{4} \) | 2.23.b_as | |
| 29 | $D_{4}$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) | 2.29.j_bo | |
| 31 | $D_{4}$ | \( 1 - 5 T + 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) | 2.31.af_cc | |
| 37 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | 2.37.e_as | |
| 41 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.41.ab_y | |
| 43 | $D_{4}$ | \( 1 - T - 38 T^{2} - p T^{3} + p^{2} T^{4} \) | 2.43.ab_abm | |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) | 2.47.ak_eg | |
| 53 | $D_{4}$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | 2.53.ad_bo | |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) | 2.59.a_cs | |
| 61 | $D_{4}$ | \( 1 + 15 T + 132 T^{2} + 15 p T^{3} + p^{2} T^{4} \) | 2.61.p_fc | |
| 67 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.67.ae_co | |
| 71 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) | 2.71.ai_bu | |
| 73 | $D_{4}$ | \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | 2.73.g_ek | |
| 79 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) | 2.79.a_du | |
| 83 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) | 2.83.aq_fu | |
| 89 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) | 2.89.q_gs | |
| 97 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | 2.97.ae_fu | |
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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
−16.8473292947, −16.4556184439, −15.6807975858, −15.2836989777, −14.9234201756, −14.5046209034, −13.8691649464, −13.4381525226, −12.5097568494, −12.2257064786, −11.9638837862, −11.4015555406, −10.8155216924, −9.86726009028, −9.52308285391, −9.01682856506, −8.46566722300, −7.63303028956, −7.32255497198, −6.45781626909, −5.82157369982, −4.98558127776, −4.03601459424, −3.62475947716, −2.61548592535, 0, 2.61548592535, 3.62475947716, 4.03601459424, 4.98558127776, 5.82157369982, 6.45781626909, 7.32255497198, 7.63303028956, 8.46566722300, 9.01682856506, 9.52308285391, 9.86726009028, 10.8155216924, 11.4015555406, 11.9638837862, 12.2257064786, 12.5097568494, 13.4381525226, 13.8691649464, 14.5046209034, 14.9234201756, 15.2836989777, 15.6807975858, 16.4556184439, 16.8473292947