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Properties

Label 2-312-104.77-c3-0-10
Degree $2$
Conductor $312$
Sign $-0.399 - 0.916i$
Analytic cond. $18.4085$
Root an. cond. $4.29052$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.32 − 1.60i)2-s − 3i·3-s + (2.83 + 7.47i)4-s + 6.73·5-s + (−4.81 + 6.98i)6-s + 30.2i·7-s + (5.40 − 21.9i)8-s − 9·9-s + (−15.6 − 10.8i)10-s + 17.0·11-s + (22.4 − 8.51i)12-s + (28.4 + 37.2i)13-s + (48.6 − 70.4i)14-s − 20.1i·15-s + (−47.8 + 42.4i)16-s − 138.·17-s + ⋯
L(s)  = 1  + (−0.823 − 0.567i)2-s − 0.577i·3-s + (0.354 + 0.934i)4-s + 0.602·5-s + (−0.327 + 0.475i)6-s + 1.63i·7-s + (0.238 − 0.971i)8-s − 0.333·9-s + (−0.495 − 0.341i)10-s + 0.468·11-s + (0.539 − 0.204i)12-s + (0.606 + 0.795i)13-s + (0.928 − 1.34i)14-s − 0.347i·15-s + (−0.748 + 0.663i)16-s − 1.96·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.399 - 0.916i$
Analytic conductor: \(18.4085\)
Root analytic conductor: \(4.29052\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :3/2),\ -0.399 - 0.916i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4701415910\)
\(L(\frac12)\) \(\approx\) \(0.4701415910\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.32 + 1.60i)T \)
3 \( 1 + 3iT \)
13 \( 1 + (-28.4 - 37.2i)T \)
good5 \( 1 - 6.73T + 125T^{2} \)
7 \( 1 - 30.2iT - 343T^{2} \)
11 \( 1 - 17.0T + 1.33e3T^{2} \)
17 \( 1 + 138.T + 4.91e3T^{2} \)
19 \( 1 + 72.5T + 6.85e3T^{2} \)
23 \( 1 + 90.4T + 1.21e4T^{2} \)
29 \( 1 + 265. iT - 2.43e4T^{2} \)
31 \( 1 + 37.6iT - 2.97e4T^{2} \)
37 \( 1 + 274.T + 5.06e4T^{2} \)
41 \( 1 - 70.0iT - 6.89e4T^{2} \)
43 \( 1 + 273. iT - 7.95e4T^{2} \)
47 \( 1 - 419. iT - 1.03e5T^{2} \)
53 \( 1 - 426. iT - 1.48e5T^{2} \)
59 \( 1 + 162.T + 2.05e5T^{2} \)
61 \( 1 - 629. iT - 2.26e5T^{2} \)
67 \( 1 - 180.T + 3.00e5T^{2} \)
71 \( 1 + 690. iT - 3.57e5T^{2} \)
73 \( 1 + 63.7iT - 3.89e5T^{2} \)
79 \( 1 - 27.2T + 4.93e5T^{2} \)
83 \( 1 - 1.09e3T + 5.71e5T^{2} \)
89 \( 1 - 1.26e3iT - 7.04e5T^{2} \)
97 \( 1 - 580. iT - 9.12e5T^{2} \)
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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

−11.66546009052174832122447032299, −10.68338861840674003490675657339, −9.297347039596442938961608800981, −8.966015042821321267387107723453, −8.070748946384714455117533109128, −6.55495783560418444990377119623, −6.05971802256525659347150118444, −4.18530655723606770343676873690, −2.35671334810114944794946861767, −1.91056232001491363005832838818, 0.20877153900720032172683168041, 1.78273431053852773053469524239, 3.80069290010843514383506780625, 4.96831484530880782434563315883, 6.31678301007848125818037689269, 6.97307909955135056042237750966, 8.248635084910105236312057342725, 9.034915681055426192359350825605, 10.12488737669841436052839503809, 10.60728232013908109546068886904

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