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Properties

Label 2-525-105.104-c1-0-22
Degree $2$
Conductor $525$
Sign $0.968 + 0.247i$
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  − 1.73·2-s + 1.73·3-s + 0.999·4-s − 2.99·6-s + (1.73 − 2i)7-s + 1.73·8-s + 2.99·9-s + 3.46i·11-s + 1.73·12-s + (−2.99 + 3.46i)14-s − 5·16-s − 6i·17-s − 5.19·18-s + 3.46i·19-s + (2.99 − 3.46i)21-s − 5.99i·22-s + ⋯
L(s)  = 1  − 1.22·2-s + 1.00·3-s + 0.499·4-s − 1.22·6-s + (0.654 − 0.755i)7-s + 0.612·8-s + 0.999·9-s + 1.04i·11-s + 0.500·12-s + (−0.801 + 0.925i)14-s − 1.25·16-s − 1.45i·17-s − 1.22·18-s + 0.794i·19-s + (0.654 − 0.755i)21-s − 1.27i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.968 + 0.247i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.968 + 0.247i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15773 - 0.145520i\)
\(L(\frac12)\) \(\approx\) \(1.15773 - 0.145520i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
5 \( 1 \)
7 \( 1 + (-1.73 + 2i)T \)
good2 \( 1 + 1.73T + 2T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 6.92T + 97T^{2} \)
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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

−10.40335084660920893921396217859, −9.770338957526745238007914661872, −9.128940917908522275627797698986, −8.148946026181638605974642358321, −7.50528039832052153106196428743, −6.95027573315165821563669462409, −4.89855763857709975144732793745, −4.07421525887621953210749049990, −2.42076348425047111622674158741, −1.18870888224860901074469397589, 1.34277948835798500127276589344, 2.57606205125438410781092667339, 3.97900958360396786414068742996, 5.27944241422446973205677357206, 6.70267301208308914027839350482, 7.76705470977258941458935466993, 8.553375506640330609193478756077, 8.800401873638799884185958784555, 9.734371329967574109255276923591, 10.74468419557150499902276473547

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