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Properties

Label 2-525-15.2-c1-0-25
Degree $2$
Conductor $525$
Sign $0.792 + 0.609i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
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  + (0.800 − 0.800i)2-s + (1.34 − 1.09i)3-s + 0.718i·4-s + (0.199 − 1.95i)6-s + (0.707 + 0.707i)7-s + (2.17 + 2.17i)8-s + (0.606 − 2.93i)9-s + 5.20i·11-s + (0.785 + 0.964i)12-s + (3.24 − 3.24i)13-s + 1.13·14-s + 2.04·16-s + (0.844 − 0.844i)17-s + (−1.86 − 2.83i)18-s − 1.32i·19-s + ⋯
L(s)  = 1  + (0.566 − 0.566i)2-s + (0.775 − 0.631i)3-s + 0.359i·4-s + (0.0813 − 0.796i)6-s + (0.267 + 0.267i)7-s + (0.769 + 0.769i)8-s + (0.202 − 0.979i)9-s + 1.56i·11-s + (0.226 + 0.278i)12-s + (0.900 − 0.900i)13-s + 0.302·14-s + 0.511·16-s + (0.204 − 0.204i)17-s + (−0.439 − 0.668i)18-s − 0.302i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.792 + 0.609i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.792 + 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47533 - 0.841445i\)
\(L(\frac12)\) \(\approx\) \(2.47533 - 0.841445i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.34 + 1.09i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-0.800 + 0.800i)T - 2iT^{2} \)
11 \( 1 - 5.20iT - 11T^{2} \)
13 \( 1 + (-3.24 + 3.24i)T - 13iT^{2} \)
17 \( 1 + (-0.844 + 0.844i)T - 17iT^{2} \)
19 \( 1 + 1.32iT - 19T^{2} \)
23 \( 1 + (5.62 + 5.62i)T + 23iT^{2} \)
29 \( 1 + 4.38T + 29T^{2} \)
31 \( 1 + 1.70T + 31T^{2} \)
37 \( 1 + (-1.71 - 1.71i)T + 37iT^{2} \)
41 \( 1 - 1.82iT - 41T^{2} \)
43 \( 1 + (-0.281 + 0.281i)T - 43iT^{2} \)
47 \( 1 + (3.39 - 3.39i)T - 47iT^{2} \)
53 \( 1 + (-3.51 - 3.51i)T + 53iT^{2} \)
59 \( 1 + 1.81T + 59T^{2} \)
61 \( 1 + 2.47T + 61T^{2} \)
67 \( 1 + (7.92 + 7.92i)T + 67iT^{2} \)
71 \( 1 + 9.06iT - 71T^{2} \)
73 \( 1 + (-1.33 + 1.33i)T - 73iT^{2} \)
79 \( 1 - 11.5iT - 79T^{2} \)
83 \( 1 + (5.46 + 5.46i)T + 83iT^{2} \)
89 \( 1 - 9.43T + 89T^{2} \)
97 \( 1 + (-3.06 - 3.06i)T + 97iT^{2} \)
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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

−10.93053947244279752079617786342, −9.931627239946769293042302259917, −8.860934786305970789250653330993, −7.973504559639976286021565672161, −7.41997635064764723211050645721, −6.18735153748007605950658840062, −4.78148784028142059235618683352, −3.81377150413563260285002952699, −2.70454402142320672237116043978, −1.76188999677948735560506003032, 1.64503526501902550920110844079, 3.55177002571541868316778655178, 4.11677888459488984507473618273, 5.43877277382276287274814859269, 6.09391864537988335359899365816, 7.36023282869682159505065473391, 8.275340144163189013140717974131, 9.123110644047595747570004540142, 10.04175385476695537685186061151, 10.89116010434743820333002973926

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