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Properties

Label 2-504-63.5-c1-0-10
Degree $2$
Conductor $504$
Sign $0.938 - 0.344i$
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  + (1.38 + 1.04i)3-s + (−0.537 − 0.930i)5-s + (−1.37 − 2.25i)7-s + (0.828 + 2.88i)9-s + (3.55 + 2.04i)11-s + (3.69 + 2.13i)13-s + (0.226 − 1.84i)15-s + (−0.717 − 1.24i)17-s + (6.41 + 3.70i)19-s + (0.446 − 4.56i)21-s + (5.43 − 3.13i)23-s + (1.92 − 3.32i)25-s + (−1.85 + 4.85i)27-s + (−8.09 + 4.67i)29-s − 6.88i·31-s + ⋯
L(s)  = 1  + (0.798 + 0.601i)3-s + (−0.240 − 0.416i)5-s + (−0.520 − 0.853i)7-s + (0.276 + 0.961i)9-s + (1.07 + 0.617i)11-s + (1.02 + 0.592i)13-s + (0.0584 − 0.477i)15-s + (−0.174 − 0.301i)17-s + (1.47 + 0.850i)19-s + (0.0975 − 0.995i)21-s + (1.13 − 0.654i)23-s + (0.384 − 0.665i)25-s + (−0.357 + 0.933i)27-s + (−1.50 + 0.868i)29-s − 1.23i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.938 - 0.344i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.938 - 0.344i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84049 + 0.327098i\)
\(L(\frac12)\) \(\approx\) \(1.84049 + 0.327098i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 - 1.04i)T \)
7 \( 1 + (1.37 + 2.25i)T \)
good5 \( 1 + (0.537 + 0.930i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.55 - 2.04i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.69 - 2.13i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.717 + 1.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.41 - 3.70i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.43 + 3.13i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.09 - 4.67i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.88iT - 31T^{2} \)
37 \( 1 + (0.453 - 0.785i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.88 - 6.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.32 + 10.9i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.43T + 47T^{2} \)
53 \( 1 + (1.50 - 0.869i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.10T + 59T^{2} \)
61 \( 1 + 2.72iT - 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 0.783iT - 71T^{2} \)
73 \( 1 + (1.95 - 1.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.63T + 79T^{2} \)
83 \( 1 + (4.48 + 7.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.71 - 2.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.05 - 2.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.88974273811857406366136862981, −9.819580970880471439331821550242, −9.310306424775490713837973187671, −8.442318614836951185949420578523, −7.39258418042140938240058749484, −6.56659545399024215542399173973, −5.01507956714787599611936320065, −4.01312979381529878161423185289, −3.36121273095587646873153172227, −1.48935386140435206770500049932, 1.37232315629021140202158011560, 3.13317356296702968906721408358, 3.47471409817780616056133997500, 5.42121206800487063953359456140, 6.45483470437949332457904910879, 7.17367991293936723056278709548, 8.294047107260338647588725029692, 9.060602789594801240610902329300, 9.612533046693991967251134331817, 11.18521970804716872695213658700

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