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Properties

Label 2-1512-9.4-c1-0-8
Degree $2$
Conductor $1512$
Sign $0.847 - 0.530i$
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  + (0.468 − 0.811i)5-s + (−0.5 − 0.866i)7-s + (2.48 + 4.30i)11-s + (−0.622 + 1.07i)13-s − 5.22·17-s + 5.18·19-s + (−1.00 + 1.73i)23-s + (2.06 + 3.57i)25-s + (3.43 + 5.95i)29-s + (2.86 − 4.96i)31-s − 0.936·35-s + 9.73·37-s + (−5.73 + 9.93i)41-s + (−4.80 − 8.31i)43-s + (−0.984 − 1.70i)47-s + ⋯
L(s)  = 1  + (0.209 − 0.362i)5-s + (−0.188 − 0.327i)7-s + (0.749 + 1.29i)11-s + (−0.172 + 0.298i)13-s − 1.26·17-s + 1.18·19-s + (−0.209 + 0.362i)23-s + (0.412 + 0.714i)25-s + (0.638 + 1.10i)29-s + (0.514 − 0.891i)31-s − 0.158·35-s + 1.59·37-s + (−0.895 + 1.55i)41-s + (−0.732 − 1.26i)43-s + (−0.143 − 0.248i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.847 - 0.530i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.847 - 0.530i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.708024389\)
\(L(\frac12)\) \(\approx\) \(1.708024389\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.468 + 0.811i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.48 - 4.30i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.622 - 1.07i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.22T + 17T^{2} \)
19 \( 1 - 5.18T + 19T^{2} \)
23 \( 1 + (1.00 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.43 - 5.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.86 + 4.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.73T + 37T^{2} \)
41 \( 1 + (5.73 - 9.93i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.80 + 8.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.984 + 1.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.63T + 53T^{2} \)
59 \( 1 + (-2.43 + 4.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.52 - 2.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.573 + 0.994i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.83T + 71T^{2} \)
73 \( 1 - 6.10T + 73T^{2} \)
79 \( 1 + (-6.05 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.431 + 0.747i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (3.78 + 6.55i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.573316982286201455747000643435, −8.928764385111143553635580367660, −7.905816748288364066892018461564, −6.97046463647150549283342725249, −6.55091947615378570041072472047, −5.23512663453737692081622393557, −4.56989411591478251716037249628, −3.64905913344268348597026754699, −2.32272371405387009023909993638, −1.20574431645130938130159315764, 0.78599749546763811495601008518, 2.40652514341558369044263041208, 3.22568286629598781079894164794, 4.29823388869312648269855971711, 5.35691597851772003676891873339, 6.32304634397377853303127442268, 6.68999733835832769935976752948, 7.952924593307225243710296331546, 8.620577943987157175886525651615, 9.361610171112712972186416768119

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