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Properties

Label 2-504-63.25-c1-0-22
Degree $2$
Conductor $504$
Sign $-0.858 + 0.513i$
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  + (−1.46 − 0.918i)3-s + (1.89 − 3.29i)5-s + (−0.841 − 2.50i)7-s + (1.31 + 2.69i)9-s + (−2.25 − 3.90i)11-s + (0.588 + 1.01i)13-s + (−5.81 + 3.08i)15-s + (−2.95 + 5.12i)17-s + (2.55 + 4.42i)19-s + (−1.06 + 4.45i)21-s + (2.09 − 3.62i)23-s + (−4.71 − 8.17i)25-s + (0.545 − 5.16i)27-s + (2.11 − 3.65i)29-s − 6.24·31-s + ⋯
L(s)  = 1  + (−0.847 − 0.530i)3-s + (0.849 − 1.47i)5-s + (−0.318 − 0.948i)7-s + (0.438 + 0.898i)9-s + (−0.680 − 1.17i)11-s + (0.163 + 0.282i)13-s + (−1.50 + 0.797i)15-s + (−0.717 + 1.24i)17-s + (0.586 + 1.01i)19-s + (−0.232 + 0.972i)21-s + (0.435 − 0.755i)23-s + (−0.943 − 1.63i)25-s + (0.105 − 0.994i)27-s + (0.392 − 0.679i)29-s − 1.12·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.858 + 0.513i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.858 + 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.256394 - 0.927659i\)
\(L(\frac12)\) \(\approx\) \(0.256394 - 0.927659i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.46 + 0.918i)T \)
7 \( 1 + (0.841 + 2.50i)T \)
good5 \( 1 + (-1.89 + 3.29i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.25 + 3.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.588 - 1.01i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.95 - 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.55 - 4.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.09 + 3.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.11 + 3.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.24T + 31T^{2} \)
37 \( 1 + (3.87 + 6.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.754 - 1.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.01 - 8.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.23T + 47T^{2} \)
53 \( 1 + (-6.49 + 11.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 1.45T + 61T^{2} \)
67 \( 1 - 1.62T + 67T^{2} \)
71 \( 1 - 8.48T + 71T^{2} \)
73 \( 1 + (-3.72 + 6.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.84T + 79T^{2} \)
83 \( 1 + (0.307 - 0.532i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.25 + 2.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.36 + 4.10i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.57170801356381066213110176945, −9.831626340223582957293661288150, −8.619735653908130520809689775935, −7.975024225884080712432448896911, −6.62488438139518663310692202014, −5.84007215982114759031491404254, −5.08358739533637097037503321521, −3.92578745431113196355229502689, −1.82493469270469666661451544293, −0.63053606578509201540225046342, 2.30594671670338875679419479834, 3.25446217318052036799692099706, 5.03805846903095670176842936569, 5.55670450609256917277198251671, 6.88199759112791641895925435281, 7.05704519084552430378320446981, 9.035842517184290228542345538358, 9.715724558152583405604057800678, 10.37362943562517112720176323536, 11.19038602267647293384096678985

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