Properties

Label 2-504-9.7-c1-0-1
Degree $2$
Conductor $504$
Sign $0.809 - 0.587i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (−1.60 − 0.649i)3-s + (−0.468 − 0.811i)5-s + (−0.5 + 0.866i)7-s + (2.15 + 2.08i)9-s + (−2.48 + 4.30i)11-s + (−0.622 − 1.07i)13-s + (0.225 + 1.60i)15-s + 5.22·17-s + 5.18·19-s + (1.36 − 1.06i)21-s + (1.00 + 1.73i)23-s + (2.06 − 3.57i)25-s + (−2.10 − 4.74i)27-s + (−3.43 + 5.95i)29-s + (2.86 + 4.96i)31-s + ⋯
L(s)  = 1  + (−0.927 − 0.374i)3-s + (−0.209 − 0.362i)5-s + (−0.188 + 0.327i)7-s + (0.718 + 0.695i)9-s + (−0.749 + 1.29i)11-s + (−0.172 − 0.298i)13-s + (0.0581 + 0.414i)15-s + 1.26·17-s + 1.18·19-s + (0.297 − 0.232i)21-s + (0.209 + 0.362i)23-s + (0.412 − 0.714i)25-s + (−0.405 − 0.913i)27-s + (−0.638 + 1.10i)29-s + (0.514 + 0.891i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.809 - 0.587i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.809 - 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.864332 + 0.280496i\)
\(L(\frac12)\) \(\approx\) \(0.864332 + 0.280496i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 + 0.649i)T \)
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

−11.11997856137327640687295841088, −10.06385262268491455063988869043, −9.543428556774815965018260087600, −7.957376836209568494920106751855, −7.48437077644404838829719437718, −6.35192622700536072449739437061, −5.25665507590211623045680511381, −4.70919705391989326264456501014, −2.95453699176211040761489115804, −1.28060752794301789759996441900, 0.71719343507070073935222794465, 3.03041912759893241808004145792, 4.04372120918891461373073898039, 5.39524394582649719854872162400, 5.95816277829176321517181297383, 7.18813561378068043131444912141, 7.932809062485240784086995649381, 9.335398170746424548974934128302, 10.06210197885271285757823444099, 10.96062514816282333194666979711

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