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Properties

Label 2-1512-8.5-c1-0-76
Degree $2$
Conductor $1512$
Sign $-0.943 + 0.332i$
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  + (−1.29 − 0.564i)2-s + (1.36 + 1.46i)4-s + 1.53i·5-s + 7-s + (−0.939 − 2.66i)8-s + (0.866 − 1.98i)10-s − 2.28i·11-s − 7.10i·13-s + (−1.29 − 0.564i)14-s + (−0.288 + 3.98i)16-s − 6.81·17-s + 1.60i·19-s + (−2.24 + 2.08i)20-s + (−1.28 + 2.96i)22-s − 1.16·23-s + ⋯
L(s)  = 1  + (−0.916 − 0.399i)2-s + (0.681 + 0.732i)4-s + 0.685i·5-s + 0.377·7-s + (−0.332 − 0.943i)8-s + (0.273 − 0.628i)10-s − 0.688i·11-s − 1.97i·13-s + (−0.346 − 0.150i)14-s + (−0.0720 + 0.997i)16-s − 1.65·17-s + 0.368i·19-s + (−0.502 + 0.467i)20-s + (−0.274 + 0.631i)22-s − 0.243·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.943 + 0.332i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.943 + 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3327777856\)
\(L(\frac12)\) \(\approx\) \(0.3327777856\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.564i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 1.53iT - 5T^{2} \)
11 \( 1 + 2.28iT - 11T^{2} \)
13 \( 1 + 7.10iT - 13T^{2} \)
17 \( 1 + 6.81T + 17T^{2} \)
19 \( 1 - 1.60iT - 19T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 - 4.07iT - 29T^{2} \)
31 \( 1 + 7.90T + 31T^{2} \)
37 \( 1 - 7.04iT - 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 + 0.344iT - 43T^{2} \)
47 \( 1 + 3.10T + 47T^{2} \)
53 \( 1 + 5.66iT - 53T^{2} \)
59 \( 1 + 1.38iT - 59T^{2} \)
61 \( 1 + 5.50iT - 61T^{2} \)
67 \( 1 + 8.35iT - 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + 5.95T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + 14.7iT - 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 2.60T + 97T^{2} \)
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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

−8.962676656911260174011424704885, −8.436105562925178449889774757335, −7.68796121208783642519113626915, −6.86804538064110462571570538707, −6.07020093603757723750988604641, −4.96345085216446537699973978763, −3.49557252890547718929802275422, −2.93784607407311011629846318967, −1.71785958790672397582566307939, −0.16875047774895437583037095398, 1.59705029580251416822873566496, 2.27288872394666746425266504607, 4.23733681011047360663074272768, 4.80300242824238037985575468622, 5.92417221854424789080843244129, 6.93346571864766816938099707446, 7.25868266660810956560729010933, 8.567905901953293879780218445252, 8.923594730582173069375920012513, 9.504548464272995386249059563859

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