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Properties

Label 2-504-63.38-c1-0-19
Degree $2$
Conductor $504$
Sign $0.910 + 0.412i$
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  + (1.72 + 0.109i)3-s + (2.05 − 3.56i)5-s + (1.89 + 1.85i)7-s + (2.97 + 0.377i)9-s + (−5.04 + 2.91i)11-s + (2.52 − 1.45i)13-s + (3.94 − 5.93i)15-s + (−1.58 + 2.73i)17-s + (0.722 − 0.417i)19-s + (3.06 + 3.40i)21-s + (−6.14 − 3.54i)23-s + (−5.96 − 10.3i)25-s + (5.10 + 0.978i)27-s + (−1.91 − 1.10i)29-s + 4.23i·31-s + ⋯
L(s)  = 1  + (0.998 + 0.0631i)3-s + (0.920 − 1.59i)5-s + (0.714 + 0.699i)7-s + (0.992 + 0.125i)9-s + (−1.52 + 0.877i)11-s + (0.699 − 0.403i)13-s + (1.01 − 1.53i)15-s + (−0.383 + 0.664i)17-s + (0.165 − 0.0956i)19-s + (0.668 + 0.743i)21-s + (−1.28 − 0.739i)23-s + (−1.19 − 2.06i)25-s + (0.982 + 0.188i)27-s + (−0.355 − 0.205i)29-s + 0.760i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26059 - 0.488100i\)
\(L(\frac12)\) \(\approx\) \(2.26059 - 0.488100i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.109i)T \)
7 \( 1 + (-1.89 - 1.85i)T \)
good5 \( 1 + (-2.05 + 3.56i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.04 - 2.91i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.52 + 1.45i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.58 - 2.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.722 + 0.417i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.14 + 3.54i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.91 + 1.10i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.23iT - 31T^{2} \)
37 \( 1 + (-1.82 - 3.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.04 + 3.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.155 + 0.269i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.00T + 47T^{2} \)
53 \( 1 + (-1.94 - 1.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 5.03T + 59T^{2} \)
61 \( 1 - 4.60iT - 61T^{2} \)
67 \( 1 + 9.99T + 67T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (3.04 + 1.76i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.15T + 79T^{2} \)
83 \( 1 + (-7.57 + 13.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.82 + 8.35i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.06 + 2.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.46598609304771220920155982678, −9.910594522551284324838838111578, −8.786929693453283839041789337524, −8.462498169548113822189804993010, −7.65525461567123061754017846703, −5.95517347607627425163134281692, −5.08516965244985440412603892188, −4.31103251483087865996476020798, −2.45472639405042766168615627603, −1.64257371079474550222753376549, 1.93824027919589581830760894049, 2.88455470717630884177063342861, 3.88443851457002902667675961355, 5.47137294726126112340267562214, 6.52847965948510594904618142979, 7.51996006963650935233208287813, 8.051899153020085225024997928840, 9.330736256630236850124354016009, 10.18633184683471525385204675494, 10.78681374986618079309814103307

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