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Properties

Label 2-504-63.58-c1-0-8
Degree $2$
Conductor $504$
Sign $0.512 - 0.858i$
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  + (1.64 + 0.553i)3-s + (−0.263 − 0.455i)5-s + (−0.333 + 2.62i)7-s + (2.38 + 1.81i)9-s + (−2.30 + 3.99i)11-s + (0.244 − 0.423i)13-s + (−0.179 − 0.893i)15-s + (2.75 + 4.77i)17-s + (1.83 − 3.18i)19-s + (−1.99 + 4.12i)21-s + (0.0269 + 0.0467i)23-s + (2.36 − 4.09i)25-s + (2.91 + 4.30i)27-s + (−3.28 − 5.68i)29-s + 6.07·31-s + ⋯
L(s)  = 1  + (0.947 + 0.319i)3-s + (−0.117 − 0.203i)5-s + (−0.125 + 0.992i)7-s + (0.796 + 0.605i)9-s + (−0.695 + 1.20i)11-s + (0.0678 − 0.117i)13-s + (−0.0464 − 0.230i)15-s + (0.668 + 1.15i)17-s + (0.421 − 0.730i)19-s + (−0.436 + 0.899i)21-s + (0.00562 + 0.00974i)23-s + (0.472 − 0.818i)25-s + (0.561 + 0.827i)27-s + (−0.609 − 1.05i)29-s + 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.512 - 0.858i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.512 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61489 + 0.916547i\)
\(L(\frac12)\) \(\approx\) \(1.61489 + 0.916547i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 0.553i)T \)
7 \( 1 + (0.333 - 2.62i)T \)
good5 \( 1 + (0.263 + 0.455i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.30 - 3.99i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.244 + 0.423i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.75 - 4.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.83 + 3.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0269 - 0.0467i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.28 + 5.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.07T + 31T^{2} \)
37 \( 1 + (-0.223 + 0.387i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.52 + 4.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.84 - 4.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.19T + 47T^{2} \)
53 \( 1 + (4.37 + 7.57i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.63T + 59T^{2} \)
61 \( 1 + 0.465T + 61T^{2} \)
67 \( 1 - 5.19T + 67T^{2} \)
71 \( 1 - 1.76T + 71T^{2} \)
73 \( 1 + (5.23 + 9.07i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 + (-4.49 - 7.78i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.05 - 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.22 - 9.04i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.89932616969772813081863731249, −9.898877754166843009165887999144, −9.403959176327348677324743149867, −8.271345227630379970710878007697, −7.82850859399769122050460954672, −6.52463701700576135250120352129, −5.23900951373941416975179039862, −4.33772690818246011275120367965, −3.00018800215054279687183016042, −2.01063493040734743396112598393, 1.10497819949197137355899448607, 2.96256578613130513993497855082, 3.57903686788314929630532667700, 5.00433685450599360009479533894, 6.35368346424915258121199813881, 7.42401393520945085037156292844, 7.88310473887077850441947283863, 8.957051380181308133105169389522, 9.843741780181742682700760718466, 10.68377164577002548005240550357

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