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Properties

Label 2-504-56.27-c1-0-1
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

\(L(1)\) \(\approx\) \(0.174192 + 0.174192i\)
\(L(\frac12)\) \(\approx\) \(0.174192 + 0.174192i\)
\(L(\frac{3}{2})\) 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

−11.05540316521488253648760653722, −10.18846182665296598781669745452, −9.670175532179177338464047724859, −8.714927077841714706904153077586, −7.70847525422549412785135462928, −6.45382725006597471321306767225, −5.35245870528907269225975476162, −4.16064833851770811675492977775, −3.17967664538115002665494991319, −1.93993926995773831038029751614, 0.13996510914528852140124123685, 2.76693032252624515375797983324, 4.11110637263828032278611242875, 5.21959637345286850389778043082, 6.07211144745850299516812234115, 7.32435750461841954403926244833, 7.49009539013041866720969105480, 9.139807199884275143500667464065, 9.483869493440824036055453644821, 10.33623733065713174767087094344

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