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Properties

Label 2-2475-165.98-c0-0-1
Degree $2$
Conductor $2475$
Sign $0.391 - 0.920i$
Analytic cond. $1.23518$
Root an. cond. $1.11138$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.541 − 0.541i)2-s + 0.414i·4-s + (−0.541 − 0.541i)7-s + (0.765 + 0.765i)8-s + i·11-s + (−1.30 + 1.30i)13-s − 0.585·14-s + 0.414·16-s + (−1.30 + 1.30i)17-s + (0.541 + 0.541i)22-s + 1.41i·26-s + (0.224 − 0.224i)28-s + 1.41·31-s + (−0.541 + 0.541i)32-s + 1.41i·34-s + ⋯
L(s)  = 1  + (0.541 − 0.541i)2-s + 0.414i·4-s + (−0.541 − 0.541i)7-s + (0.765 + 0.765i)8-s + i·11-s + (−1.30 + 1.30i)13-s − 0.585·14-s + 0.414·16-s + (−1.30 + 1.30i)17-s + (0.541 + 0.541i)22-s + 1.41i·26-s + (0.224 − 0.224i)28-s + 1.41·31-s + (−0.541 + 0.541i)32-s + 1.41i·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.391 - 0.920i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ 0.391 - 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.228118824\)
\(L(\frac12)\) \(\approx\) \(1.228118824\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 - iT \)
good2 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
7 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
13 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
17 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.318607041038915346724331754453, −8.550135584124540761017121342691, −7.56622051000768432409748189816, −6.99411852605332968260145318881, −6.30916767248735465938902269118, −4.88967087757416835302463748468, −4.36444838962557859037199078207, −3.77538194145660004783146234185, −2.49393099850133096032803454770, −1.92110933439714093409463164729, 0.64994610470216955402587364120, 2.48206958166715226537219666954, 3.14617270754965061723162289905, 4.55335709899731559483648704156, 5.03816915291608886732158909243, 5.95499655965487718103446889134, 6.40559050543656492582610161580, 7.34784372829716523790681459236, 8.007341752759602259584494150137, 9.137108622304505269227031686154

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