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Properties

Label 2-312-104.19-c1-0-21
Degree $2$
Conductor $312$
Sign $0.280 + 0.959i$
Analytic cond. $2.49133$
Root an. cond. $1.57839$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.21 − 0.722i)2-s + (0.5 − 0.866i)3-s + (0.957 − 1.75i)4-s + (−0.403 − 0.403i)5-s + (−0.0173 − 1.41i)6-s + (4.65 + 1.24i)7-s + (−0.103 − 2.82i)8-s + (−0.499 − 0.866i)9-s + (−0.782 − 0.199i)10-s + (−5.72 + 1.53i)11-s + (−1.04 − 1.70i)12-s + (−0.849 + 3.50i)13-s + (6.55 − 1.84i)14-s + (−0.551 + 0.147i)15-s + (−2.16 − 3.36i)16-s + (−0.917 + 0.529i)17-s + ⋯
L(s)  = 1  + (0.859 − 0.510i)2-s + (0.288 − 0.499i)3-s + (0.478 − 0.878i)4-s + (−0.180 − 0.180i)5-s + (−0.00707 − 0.577i)6-s + (1.75 + 0.471i)7-s + (−0.0367 − 0.999i)8-s + (−0.166 − 0.288i)9-s + (−0.247 − 0.0630i)10-s + (−1.72 + 0.462i)11-s + (−0.300 − 0.492i)12-s + (−0.235 + 0.971i)13-s + (1.75 − 0.492i)14-s + (−0.142 + 0.0381i)15-s + (−0.541 − 0.840i)16-s + (−0.222 + 0.128i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.280 + 0.959i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ 0.280 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87433 - 1.40572i\)
\(L(\frac12)\) \(\approx\) \(1.87433 - 1.40572i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.21 + 0.722i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.849 - 3.50i)T \)
good5 \( 1 + (0.403 + 0.403i)T + 5iT^{2} \)
7 \( 1 + (-4.65 - 1.24i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (5.72 - 1.53i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.917 - 0.529i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.617 - 0.165i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.02 + 1.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.86 - 2.81i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.54 + 2.54i)T + 31iT^{2} \)
37 \( 1 + (-1.83 - 6.84i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.37 - 5.12i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (8.58 - 4.95i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.92 - 5.92i)T - 47iT^{2} \)
53 \( 1 - 1.19iT - 53T^{2} \)
59 \( 1 + (-2.11 + 7.89i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.30 + 3.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.06 + 7.71i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.113 + 0.422i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-7.34 - 7.34i)T + 73iT^{2} \)
79 \( 1 + 7.20iT - 79T^{2} \)
83 \( 1 + (-7.79 + 7.79i)T - 83iT^{2} \)
89 \( 1 + (0.708 - 0.189i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.48 + 1.20i)T + (84.0 + 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.60724817434740655909225516300, −10.92506088137151534969379597337, −9.827435878120612301959142816887, −8.417268912617238827666829132441, −7.73839109770677943779067740037, −6.46959502542741543742210119301, −5.02920980458220544177771144573, −4.62272924140806578521424537066, −2.69148212257369118143788401698, −1.72955028165285519362255284505, 2.46706692532340034561219214500, 3.73829158164811950369885511371, 5.11353850305542926006385934902, 5.33295879487859266206439775568, 7.28819024887384992833941975143, 7.914859789168105482097744261705, 8.572224232052077672045614458213, 10.41663550753948944603668816051, 10.94971158950878722184143677308, 11.80126851390103744018520781765

Graph of the $Z$-function along the critical line