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Properties

Label 2-525-15.2-c1-0-3
Degree $2$
Conductor $525$
Sign $-0.326 - 0.945i$
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  + (−1.27 + 1.27i)2-s + (−0.774 − 1.54i)3-s − 1.23i·4-s + (2.95 + 0.984i)6-s + (−0.707 − 0.707i)7-s + (−0.971 − 0.971i)8-s + (−1.79 + 2.40i)9-s + 0.596i·11-s + (−1.91 + 0.958i)12-s + (0.651 − 0.651i)13-s + 1.79·14-s + 4.94·16-s + (−2.63 + 2.63i)17-s + (−0.765 − 5.34i)18-s + 1.16i·19-s + ⋯
L(s)  = 1  + (−0.899 + 0.899i)2-s + (−0.447 − 0.894i)3-s − 0.618i·4-s + (1.20 + 0.402i)6-s + (−0.267 − 0.267i)7-s + (−0.343 − 0.343i)8-s + (−0.599 + 0.800i)9-s + 0.179i·11-s + (−0.552 + 0.276i)12-s + (0.180 − 0.180i)13-s + 0.480·14-s + 1.23·16-s + (−0.638 + 0.638i)17-s + (−0.180 − 1.25i)18-s + 0.267i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.326 - 0.945i$
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: \((2,\ 525,\ (\ :1/2),\ -0.326 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.286038 + 0.401473i\)
\(L(\frac12)\) \(\approx\) \(0.286038 + 0.401473i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.774 + 1.54i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (1.27 - 1.27i)T - 2iT^{2} \)
11 \( 1 - 0.596iT - 11T^{2} \)
13 \( 1 + (-0.651 + 0.651i)T - 13iT^{2} \)
17 \( 1 + (2.63 - 2.63i)T - 17iT^{2} \)
19 \( 1 - 1.16iT - 19T^{2} \)
23 \( 1 + (-2.83 - 2.83i)T + 23iT^{2} \)
29 \( 1 - 9.57T + 29T^{2} \)
31 \( 1 + 7.54T + 31T^{2} \)
37 \( 1 + (-7.81 - 7.81i)T + 37iT^{2} \)
41 \( 1 - 8.32iT - 41T^{2} \)
43 \( 1 + (7.71 - 7.71i)T - 43iT^{2} \)
47 \( 1 + (5.33 - 5.33i)T - 47iT^{2} \)
53 \( 1 + (-2.54 - 2.54i)T + 53iT^{2} \)
59 \( 1 - 4.37T + 59T^{2} \)
61 \( 1 + 15.0T + 61T^{2} \)
67 \( 1 + (-1.26 - 1.26i)T + 67iT^{2} \)
71 \( 1 + 7.16iT - 71T^{2} \)
73 \( 1 + (-5.66 + 5.66i)T - 73iT^{2} \)
79 \( 1 + 3.85iT - 79T^{2} \)
83 \( 1 + (-4.70 - 4.70i)T + 83iT^{2} \)
89 \( 1 - 2.19T + 89T^{2} \)
97 \( 1 + (-7.69 - 7.69i)T + 97iT^{2} \)
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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

−11.03297829468879322296308113194, −10.09225391864883133319830946355, −9.126698749890951512846129378145, −8.143405802643478577735544857437, −7.64849627703124133786764183032, −6.51290606266356970732484455782, −6.25475434744965308058800186425, −4.82610930234863356533496945668, −3.10632043145777248170979645588, −1.25643185432961103798569333212, 0.45885342307254961090150899207, 2.39275305903873476491090947720, 3.51594800960738769819263911617, 4.84596710420286643225708436488, 5.83682312220692628274745549511, 6.94458266262418416567501599648, 8.562307509589040252507713354946, 9.000071307835478478758587208527, 9.815099349107051681791238564337, 10.58867643637315001517256557663

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