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Properties

Label 2-504-9.4-c1-0-16
Degree $2$
Conductor $504$
Sign $-0.500 + 0.866i$
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  + (0.592 − 1.62i)3-s + (0.326 − 0.565i)5-s + (−0.5 − 0.866i)7-s + (−2.29 − 1.92i)9-s + (−1.70 − 2.95i)11-s + (0.152 − 0.264i)13-s + (−0.726 − 0.866i)15-s − 0.226·17-s − 2.16·19-s + (−1.70 + 0.300i)21-s + (3.35 − 5.80i)23-s + (2.28 + 3.96i)25-s + (−4.5 + 2.59i)27-s + (0.254 + 0.441i)29-s + (2.85 − 4.95i)31-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)3-s + (0.145 − 0.252i)5-s + (−0.188 − 0.327i)7-s + (−0.766 − 0.642i)9-s + (−0.514 − 0.890i)11-s + (0.0423 − 0.0733i)13-s + (−0.187 − 0.223i)15-s − 0.0549·17-s − 0.496·19-s + (−0.372 + 0.0656i)21-s + (0.698 − 1.21i)23-s + (0.457 + 0.792i)25-s + (−0.866 + 0.499i)27-s + (0.0473 + 0.0819i)29-s + (0.513 − 0.889i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.500 + 0.866i$
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: \((2,\ 504,\ (\ :1/2),\ -0.500 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678403 - 1.17502i\)
\(L(\frac12)\) \(\approx\) \(0.678403 - 1.17502i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.592 + 1.62i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.326 + 0.565i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.152 + 0.264i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.226T + 17T^{2} \)
19 \( 1 + 2.16T + 19T^{2} \)
23 \( 1 + (-3.35 + 5.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.254 - 0.441i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.85 + 4.95i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.28T + 37T^{2} \)
41 \( 1 + (0.479 - 0.829i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.85 - 6.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.14 + 7.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.18T + 53T^{2} \)
59 \( 1 + (-6.07 + 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.75 - 3.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.87 - 4.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + (-5.98 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.13 + 3.69i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.19T + 89T^{2} \)
97 \( 1 + (-7.48 - 12.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.78354899075047393165126747245, −9.606435004208453385951404148873, −8.611780588329477057639355597826, −8.042859423141881781045623486799, −6.92557219397280721987186634241, −6.17668757956916618735130427995, −5.04155971070332916073826332537, −3.52047580560689740919849233935, −2.40494799581511617562534481903, −0.77630186302711579070772874328, 2.25598511714636860449218859859, 3.34060349815947353797620413798, 4.57207570036140324699453865857, 5.39692742367434681675938061220, 6.61455204932135719750535433521, 7.72730377094166574420335477751, 8.723729978606087083311341163481, 9.504360900508523386529055350463, 10.29029315030739885064185081972, 10.94544932211986451384329182753

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