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Properties

Label 2-1512-24.11-c1-0-50
Degree $2$
Conductor $1512$
Sign $0.707 - 0.707i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + (1.36 + 0.366i)2-s + (1.73 + i)4-s − 1.12·5-s − i·7-s + (1.99 + 2i)8-s + (−1.54 − 0.413i)10-s + 3.86i·11-s − 5.08i·13-s + (0.366 − 1.36i)14-s + (1.99 + 3.46i)16-s + 2i·17-s + 7.99·19-s + (−1.95 − 1.12i)20-s + (−1.41 + 5.27i)22-s + 7.68·23-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.5i)4-s − 0.505·5-s − 0.377i·7-s + (0.707 + 0.707i)8-s + (−0.488 − 0.130i)10-s + 1.16i·11-s − 1.41i·13-s + (0.0978 − 0.365i)14-s + (0.499 + 0.866i)16-s + 0.485i·17-s + 1.83·19-s + (−0.437 − 0.252i)20-s + (−0.301 + 1.12i)22-s + 1.60·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.087901571\)
\(L(\frac12)\) \(\approx\) \(3.087901571\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 1.12T + 5T^{2} \)
11 \( 1 - 3.86iT - 11T^{2} \)
13 \( 1 + 5.08iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 7.99T + 19T^{2} \)
23 \( 1 - 7.68T + 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 - 4.08iT - 31T^{2} \)
37 \( 1 - 8.28iT - 37T^{2} \)
41 \( 1 - 8.41iT - 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 - 1.93T + 47T^{2} \)
53 \( 1 - 7.72T + 53T^{2} \)
59 \( 1 + 10.9iT - 59T^{2} \)
61 \( 1 + 11.4iT - 61T^{2} \)
67 \( 1 - 5.08T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 + 2.63T + 73T^{2} \)
79 \( 1 + 7.17iT - 79T^{2} \)
83 \( 1 + 14.9iT - 83T^{2} \)
89 \( 1 - 12.6iT - 89T^{2} \)
97 \( 1 - 4.17T + 97T^{2} \)
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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

−9.828223875926158508865446202802, −8.437117777337960570838896415822, −7.72353810115171926569197102699, −7.17322285946874360868703462483, −6.33199091610995526098759331654, −5.10914515814431260585737884721, −4.80611508928106236991808774188, −3.49332317027677041487147603625, −2.96740213633203243321329636370, −1.35538471196121924546206470057, 1.04651832671260405631779705897, 2.50418834702289209286596715828, 3.39316964897096732452893671408, 4.21040677457172613559844051990, 5.25073789592091124198759351518, 5.84467686062595664036046438431, 6.96856445409531789834215083084, 7.42980770727861736458941524874, 8.698931975815026066380791689500, 9.345022191103800518688349557127

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