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Properties

Label 2-504-56.37-c1-0-22
Degree $2$
Conductor $504$
Sign $0.752 + 0.658i$
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  + (0.189 + 1.40i)2-s + (−1.92 + 0.531i)4-s + (−3.09 − 1.78i)5-s + (0.993 + 2.45i)7-s + (−1.11 − 2.60i)8-s + (1.91 − 4.68i)10-s + (0.815 − 0.470i)11-s − 6.15i·13-s + (−3.24 + 1.85i)14-s + (3.43 − 2.05i)16-s + (−1.89 − 3.27i)17-s + (2.09 + 1.20i)19-s + (6.92 + 1.80i)20-s + (0.814 + 1.05i)22-s + (1.49 − 2.58i)23-s + ⋯
L(s)  = 1  + (0.134 + 0.990i)2-s + (−0.964 + 0.265i)4-s + (−1.38 − 0.800i)5-s + (0.375 + 0.926i)7-s + (−0.392 − 0.919i)8-s + (0.606 − 1.48i)10-s + (0.245 − 0.142i)11-s − 1.70i·13-s + (−0.868 + 0.496i)14-s + (0.858 − 0.512i)16-s + (−0.458 − 0.794i)17-s + (0.479 + 0.276i)19-s + (1.54 + 0.402i)20-s + (0.173 + 0.224i)22-s + (0.311 − 0.539i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.752 + 0.658i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.752 + 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.694476 - 0.260964i\)
\(L(\frac12)\) \(\approx\) \(0.694476 - 0.260964i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.189 - 1.40i)T \)
3 \( 1 \)
7 \( 1 + (-0.993 - 2.45i)T \)
good5 \( 1 + (3.09 + 1.78i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.815 + 0.470i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.15iT - 13T^{2} \)
17 \( 1 + (1.89 + 3.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.09 - 1.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.49 + 2.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.68iT - 29T^{2} \)
31 \( 1 + (5.35 + 9.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.47 + 0.853i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.55T + 41T^{2} \)
43 \( 1 + 3.50iT - 43T^{2} \)
47 \( 1 + (3.42 - 5.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.57 - 3.79i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.100 - 0.0580i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.06 + 4.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.44 - 1.98i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.92T + 71T^{2} \)
73 \( 1 + (3.11 + 5.39i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.73 + 4.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.19iT - 83T^{2} \)
89 \( 1 + (0.910 - 1.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−10.99130456044474831301053315642, −9.489472421967157909157682508871, −8.781528739844385804507444217540, −7.908015761794323399652649926237, −7.57550531689464140343228946536, −6.03145167114220069413229250314, −5.18880845424691983072150713649, −4.34839488072975376002372060963, −3.14714869399630146016870614549, −0.45264421063736873724365139753, 1.60567191462403454476580248330, 3.32833318251052018040384001203, 4.02264731838997863313434099991, 4.82881787636482372846918912089, 6.67187697804553369694978334345, 7.38234433965305421178985779997, 8.454315147984612109049931711843, 9.361004054765793707475566239709, 10.53825236209664114097799996092, 11.12175069447016099476293185738

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