Properties

Label 2-504-56.27-c1-0-10
Degree $2$
Conductor $504$
Sign $-i$
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  + 1.41i·2-s − 2.00·4-s + 2.64·7-s − 2.82i·8-s + 5.29·13-s + 3.74i·14-s + 4.00·16-s + 7.48i·17-s + 2.82i·23-s − 5·25-s + 7.48i·26-s − 5.29·28-s − 5.65i·29-s + 5.29·31-s + 5.65i·32-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s + 0.999·7-s − 1.00i·8-s + 1.46·13-s + 1.00i·14-s + 1.00·16-s + 1.81i·17-s + 0.589i·23-s − 25-s + 1.46i·26-s − 0.999·28-s − 1.05i·29-s + 0.950·31-s + 1.00i·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03501 + 1.03501i\)
\(L(\frac12)\) \(\approx\) \(1.03501 + 1.03501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
7 \( 1 - 2.64T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 5.29T + 13T^{2} \)
17 \( 1 - 7.48iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 - 5.29T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 7.48iT - 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 + 14.9iT - 59T^{2} \)
61 \( 1 - 5.29T + 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 14.9iT - 83T^{2} \)
89 \( 1 - 7.48iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.05217263053193508608402770534, −10.18909076219679421567632300995, −9.069979887274460566361604239553, −8.157935482570168780414913855375, −7.82542503405906086005316133827, −6.31651390568438319190576419983, −5.83755660888096284262870974688, −4.53261572760928652824758187709, −3.70548012195453083070796034945, −1.48453636019026352839449672475, 1.09404301299056981693825080023, 2.48310195227880672755422352283, 3.78271851312431008491723456974, 4.79995957462417333315078955086, 5.71887489575259466224404756806, 7.21918409230934678474744023169, 8.354020423319738704714545828063, 8.912311476293167907385932230570, 9.986092497788141681567145936132, 10.87379481034767377537910743500

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