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Properties

Label 2-504-21.17-c1-0-5
Degree $2$
Conductor $504$
Sign $0.488 + 0.872i$
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  + (−0.587 − 1.01i)5-s + (2.35 − 1.21i)7-s + (−1.44 − 0.835i)11-s − 1.14i·13-s + (2.07 − 3.59i)17-s + (1.83 − 1.05i)19-s + (−4.22 + 2.43i)23-s + (1.81 − 3.13i)25-s − 8.32i·29-s + (7.18 + 4.14i)31-s + (−2.61 − 1.68i)35-s + (−2.19 − 3.81i)37-s + 2.67·41-s + 4.08·43-s + (1.75 + 3.03i)47-s + ⋯
L(s)  = 1  + (−0.262 − 0.454i)5-s + (0.889 − 0.457i)7-s + (−0.436 − 0.251i)11-s − 0.317i·13-s + (0.503 − 0.872i)17-s + (0.420 − 0.242i)19-s + (−0.880 + 0.508i)23-s + (0.362 − 0.627i)25-s − 1.54i·29-s + (1.28 + 0.744i)31-s + (−0.441 − 0.284i)35-s + (−0.361 − 0.626i)37-s + 0.417·41-s + 0.623·43-s + (0.255 + 0.442i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.488 + 0.872i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.488 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21715 - 0.713890i\)
\(L(\frac12)\) \(\approx\) \(1.21715 - 0.713890i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.35 + 1.21i)T \)
good5 \( 1 + (0.587 + 1.01i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.44 + 0.835i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.14iT - 13T^{2} \)
17 \( 1 + (-2.07 + 3.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.83 + 1.05i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.22 - 2.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.32iT - 29T^{2} \)
31 \( 1 + (-7.18 - 4.14i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.19 + 3.81i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.67T + 41T^{2} \)
43 \( 1 - 4.08T + 43T^{2} \)
47 \( 1 + (-1.75 - 3.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.59 - 0.922i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.98 - 5.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (12.8 - 7.39i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.22 - 7.31i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.48iT - 71T^{2} \)
73 \( 1 + (-0.846 - 0.488i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.56 + 9.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (-6.29 - 10.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.52iT - 97T^{2} \)
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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

−10.74609958658687025174875301587, −9.985857655470665258446063315229, −8.892294003614920271834111707882, −7.941234323351212976789703940272, −7.45144356750098524995774295357, −5.99925520708971431062631443919, −4.99532075063548901547385148355, −4.14043001249977963403898754022, −2.66483752469251716008892550678, −0.923999583103398919400872450493, 1.72881485695157410568676563282, 3.10389771230581358285696094969, 4.40506812474486320890247469284, 5.41872667310306939957883366889, 6.46853870595772557990530786378, 7.62553759789441722400085380815, 8.230483976430313910485789994475, 9.263069573970258835831644295585, 10.37417527173221267437912825106, 10.98328447615790170881097995723

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