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Properties

Label 2-504-168.11-c1-0-2
Degree $2$
Conductor $504$
Sign $0.720 - 0.693i$
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  + (0.366 − 1.36i)2-s + (−1.73 − i)4-s + (1.91 + 3.31i)5-s + (−2.09 + 1.62i)7-s + (−2 + 1.99i)8-s + (5.22 − 1.40i)10-s + (−3.82 − 2.20i)11-s + 5.41i·13-s + (1.44 + 3.44i)14-s + (1.99 + 3.46i)16-s + (1.94 + 1.12i)17-s + (2.70 + 4.68i)19-s − 7.65i·20-s + (−4.41 + 4.41i)22-s + (−1.70 − 2.95i)23-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.5i)4-s + (0.856 + 1.48i)5-s + (−0.790 + 0.612i)7-s + (−0.707 + 0.707i)8-s + (1.65 − 0.443i)10-s + (−1.15 − 0.665i)11-s + 1.50i·13-s + (0.387 + 0.921i)14-s + (0.499 + 0.866i)16-s + (0.471 + 0.271i)17-s + (0.621 + 1.07i)19-s − 1.71i·20-s + (−0.941 + 0.941i)22-s + (−0.355 − 0.616i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.720 - 0.693i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.720 - 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14523 + 0.461505i\)
\(L(\frac12)\) \(\approx\) \(1.14523 + 0.461505i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
3 \( 1 \)
7 \( 1 + (2.09 - 1.62i)T \)
good5 \( 1 + (-1.91 - 3.31i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.82 + 2.20i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.41iT - 13T^{2} \)
17 \( 1 + (-1.94 - 1.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.70 - 4.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.70 + 2.95i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (1.37 + 0.792i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.39 + 2.53i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.17iT - 41T^{2} \)
43 \( 1 - 8.58T + 43T^{2} \)
47 \( 1 + (-2.70 - 4.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.91 + 10.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.25 - 3.03i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.94 - 1.12i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.82 - 3.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.89T + 71T^{2} \)
73 \( 1 + (7.65 - 13.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.25 - 3.03i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.07iT - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−10.97458992769048703584048389349, −10.15826105996586740745770163295, −9.686722180159022035095301194801, −8.679874487096463650421052688398, −7.27551827615902118593275944680, −6.01172827869766397113197575287, −5.68037609748578231400032013035, −3.88072744281486813196806707693, −2.86607746189045199990371475019, −2.10829705496825386771136473893, 0.67004294413124188324539306847, 2.96283566005710961424221677802, 4.46470360014473993305892644788, 5.35220671302165710260540577736, 5.85428843686856879287947638716, 7.32515551603728179908518599576, 7.900502340525525552014932001233, 9.058585078471190170237199477802, 9.691660176200755478931760862630, 10.40716809024656483612299899282

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