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Properties

Label 2-1512-7.4-c1-0-12
Degree $2$
Conductor $1512$
Sign $0.911 - 0.411i$
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  + (0.849 + 1.47i)5-s + (−2.64 − 0.0963i)7-s + (2.14 − 3.71i)11-s + 2.69·13-s + (−1.64 + 2.84i)17-s + (1.79 + 3.10i)19-s + (−1.29 − 2.24i)23-s + (1.05 − 1.82i)25-s + 7.68·29-s + (0.461 − 0.798i)31-s + (−2.10 − 3.97i)35-s + (5.23 + 9.06i)37-s − 5.76·41-s + 11.0·43-s + (−2.66 − 4.60i)47-s + ⋯
L(s)  = 1  + (0.380 + 0.658i)5-s + (−0.999 − 0.0364i)7-s + (0.646 − 1.11i)11-s + 0.748·13-s + (−0.398 + 0.690i)17-s + (0.411 + 0.712i)19-s + (−0.269 − 0.467i)23-s + (0.211 − 0.365i)25-s + 1.42·29-s + (0.0828 − 0.143i)31-s + (−0.355 − 0.671i)35-s + (0.860 + 1.48i)37-s − 0.900·41-s + 1.69·43-s + (−0.388 − 0.672i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.911 - 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.723670920\)
\(L(\frac12)\) \(\approx\) \(1.723670920\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (2.64 + 0.0963i)T \)
good5 \( 1 + (-0.849 - 1.47i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.14 + 3.71i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.69T + 13T^{2} \)
17 \( 1 + (1.64 - 2.84i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.79 - 3.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.29 + 2.24i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.68T + 29T^{2} \)
31 \( 1 + (-0.461 + 0.798i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.23 - 9.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.76T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + (2.66 + 4.60i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.23 - 3.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.60 - 7.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.08 - 7.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.81 + 3.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + (0.333 - 0.576i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.88 - 8.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.25T + 83T^{2} \)
89 \( 1 + (-0.800 - 1.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.823T + 97T^{2} \)
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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

−9.604917144899453037575071224064, −8.685042694777030680132446529635, −8.110431114378918336540003818537, −6.75142171247455393907261175612, −6.33273622773563382758207155062, −5.77658175635583806259897223075, −4.27002044421659042448702003702, −3.41350554632371410043514656150, −2.63090430460112641196767986277, −1.04602380845999204640956058979, 0.892183752963949120746116369919, 2.22398594943043149415110915199, 3.39195785126390623255384090088, 4.41973745157907860269017380624, 5.22854323198508580316783452888, 6.27895525587811457930267783923, 6.86408326804164068600005792413, 7.77213490371558004620248482348, 8.993294462500198596934050269753, 9.304835486749444650099581057089

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