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Properties

Label 2-504-9.4-c1-0-3
Degree $2$
Conductor $504$
Sign $-0.661 - 0.750i$
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  + (0.468 + 1.66i)3-s + (−0.553 + 0.958i)5-s + (0.5 + 0.866i)7-s + (−2.56 + 1.56i)9-s + (−0.433 − 0.751i)11-s + (−2.70 + 4.67i)13-s + (−1.85 − 0.473i)15-s + 5.12·17-s − 5.44·19-s + (−1.20 + 1.23i)21-s + (0.967 − 1.67i)23-s + (1.88 + 3.27i)25-s + (−3.80 − 3.53i)27-s + (−1.57 − 2.72i)29-s + (−3.02 + 5.24i)31-s + ⋯
L(s)  = 1  + (0.270 + 0.962i)3-s + (−0.247 + 0.428i)5-s + (0.188 + 0.327i)7-s + (−0.853 + 0.521i)9-s + (−0.130 − 0.226i)11-s + (−0.748 + 1.29i)13-s + (−0.479 − 0.122i)15-s + 1.24·17-s − 1.24·19-s + (−0.263 + 0.270i)21-s + (0.201 − 0.349i)23-s + (0.377 + 0.654i)25-s + (−0.732 − 0.680i)27-s + (−0.292 − 0.506i)29-s + (−0.543 + 0.941i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.661 - 0.750i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.661 - 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506059 + 1.12083i\)
\(L(\frac12)\) \(\approx\) \(0.506059 + 1.12083i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.468 - 1.66i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.553 - 0.958i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.433 + 0.751i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.70 - 4.67i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.12T + 17T^{2} \)
19 \( 1 + 5.44T + 19T^{2} \)
23 \( 1 + (-0.967 + 1.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.57 + 2.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.02 - 5.24i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.49T + 37T^{2} \)
41 \( 1 + (1.71 - 2.96i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.41 + 5.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.43 - 7.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.223T + 53T^{2} \)
59 \( 1 + (-6.30 + 10.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.46 - 7.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.17 - 5.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.33T + 71T^{2} \)
73 \( 1 - 16.2T + 73T^{2} \)
79 \( 1 + (-2.44 - 4.23i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.33 + 10.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 + (-0.761 - 1.31i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.14629328906880893322289546709, −10.34763826724707802458066137020, −9.479424794376552083989387802495, −8.719623640503659143428166521674, −7.77028957303199521014370128471, −6.66653362028995998545203279207, −5.46121140379228147152474380497, −4.50466298130891489394878019053, −3.47380363507071933368185460927, −2.28190982831572242508250141290, 0.70483203917043309216997941109, 2.29499151575770233367714213896, 3.57093636524863493917901440930, 5.00375759875484506098832351472, 5.96990949777248325876763658745, 7.18131356893267816025137995168, 7.87165938733009260535903782111, 8.509123064996112965230894055567, 9.696143920842419806141064198023, 10.60864523481030281188591423518

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