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Properties

Label 2-975-195.83-c0-0-1
Degree $2$
Conductor $975$
Sign $0.661 - 0.749i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
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  + (0.707 + 0.707i)3-s − 4-s + 1.41·7-s + 1.00i·9-s + (−0.707 − 0.707i)12-s + (−0.707 − 0.707i)13-s + 16-s + (1 + i)19-s + (1.00 + 1.00i)21-s + (−0.707 + 0.707i)27-s − 1.41·28-s + (−1 + i)31-s − 1.00i·36-s − 1.41·37-s − 1.00i·39-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s − 4-s + 1.41·7-s + 1.00i·9-s + (−0.707 − 0.707i)12-s + (−0.707 − 0.707i)13-s + 16-s + (1 + i)19-s + (1.00 + 1.00i)21-s + (−0.707 + 0.707i)27-s − 1.41·28-s + (−1 + i)31-s − 1.00i·36-s − 1.41·37-s − 1.00i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.661 - 0.749i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (668, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.661 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.155701168\)
\(L(\frac12)\) \(\approx\) \(1.155701168\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
13 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + T^{2} \)
7 \( 1 - 1.41T + T^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (1 - i)T - iT^{2} \)
37 \( 1 + 1.41T + T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41iT - T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + 1.41iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + 1.41iT - T^{2} \)
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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

−10.31686981000691905675717092829, −9.364042543691909367388319888619, −8.726335703055259599504616149829, −7.936010588213315571878748305674, −7.45231618382027600261528430987, −5.39230848813497756657406462118, −5.15889940988448803420875317195, −4.12364525615566874718355605243, −3.24114219794564339667745721942, −1.73204829025074932888958712334, 1.29970898184641827962357204357, 2.53333284196678968956587394001, 3.90541384040731490355753390387, 4.76867834752321037071170106292, 5.62136928952977858031473985736, 7.06286031157974745525103694772, 7.67184346063692864877682565366, 8.434555229370084385578975096736, 9.170536959712961041296604584554, 9.726735920442098690511687035723

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