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Properties

Label 2-312-104.77-c3-0-21
Degree $2$
Conductor $312$
Sign $0.0418 - 0.999i$
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  + (−0.800 − 2.71i)2-s + 3i·3-s + (−6.71 + 4.34i)4-s + 12.1·5-s + (8.13 − 2.40i)6-s + 21.1i·7-s + (17.1 + 14.7i)8-s − 9·9-s + (−9.70 − 32.8i)10-s + 16.0·11-s + (−13.0 − 20.1i)12-s + (−36.8 + 29.0i)13-s + (57.3 − 16.9i)14-s + 36.3i·15-s + (26.2 − 58.3i)16-s + 45.8·17-s + ⋯
L(s)  = 1  + (−0.283 − 0.959i)2-s + 0.577i·3-s + (−0.839 + 0.543i)4-s + 1.08·5-s + (0.553 − 0.163i)6-s + 1.14i·7-s + (0.758 + 0.651i)8-s − 0.333·9-s + (−0.307 − 1.04i)10-s + 0.438·11-s + (−0.313 − 0.484i)12-s + (−0.785 + 0.619i)13-s + (1.09 − 0.323i)14-s + 0.626i·15-s + (0.410 − 0.911i)16-s + 0.653·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0418 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0418 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.0418 - 0.999i$
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: \((2,\ 312,\ (\ :3/2),\ 0.0418 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.179059389\)
\(L(\frac12)\) \(\approx\) \(1.179059389\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.800 + 2.71i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (36.8 - 29.0i)T \)
good5 \( 1 - 12.1T + 125T^{2} \)
7 \( 1 - 21.1iT - 343T^{2} \)
11 \( 1 - 16.0T + 1.33e3T^{2} \)
17 \( 1 - 45.8T + 4.91e3T^{2} \)
19 \( 1 + 117.T + 6.85e3T^{2} \)
23 \( 1 + 97.3T + 1.21e4T^{2} \)
29 \( 1 + 90.0iT - 2.43e4T^{2} \)
31 \( 1 - 151. iT - 2.97e4T^{2} \)
37 \( 1 + 66.3T + 5.06e4T^{2} \)
41 \( 1 - 112. iT - 6.89e4T^{2} \)
43 \( 1 - 234. iT - 7.95e4T^{2} \)
47 \( 1 - 580. iT - 1.03e5T^{2} \)
53 \( 1 + 241. iT - 1.48e5T^{2} \)
59 \( 1 - 313.T + 2.05e5T^{2} \)
61 \( 1 + 334. iT - 2.26e5T^{2} \)
67 \( 1 + 315.T + 3.00e5T^{2} \)
71 \( 1 - 503. iT - 3.57e5T^{2} \)
73 \( 1 - 757. iT - 3.89e5T^{2} \)
79 \( 1 - 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + 822.T + 5.71e5T^{2} \)
89 \( 1 + 803. iT - 7.04e5T^{2} \)
97 \( 1 - 750. iT - 9.12e5T^{2} \)
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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

−11.47364142516654068641143758054, −10.34137745330191595083208506776, −9.651761812335574791902471634108, −9.069208956154922869011786998040, −8.121676673823242371189376214625, −6.33612064547317664158597688652, −5.31620062567400372905161179271, −4.21560060835508425519798966254, −2.69840881777282360107385461779, −1.81605293288851458172496694496, 0.46013426914740574840634622432, 1.89984237063254858986635722394, 3.98155114294725685901483561543, 5.34523757654177386543741998049, 6.24698298894562390161895918654, 7.10729348867012960225216834621, 7.934714935461441086867768220096, 9.021025111441561956215815490498, 10.10435286763500373279353155093, 10.50672997892905602528135333954

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