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Properties

Label 2-975-195.194-c0-0-1
Degree $2$
Conductor $975$
Sign $0.447 + 0.894i$
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  − i·3-s + 4-s − 9-s − i·12-s − i·13-s + 16-s + i·27-s − 36-s − 39-s + 2i·43-s − i·48-s − 49-s − i·52-s − 2·61-s + 64-s + ⋯
L(s)  = 1  − i·3-s + 4-s − 9-s − i·12-s − i·13-s + 16-s + i·27-s − 36-s − 39-s + 2i·43-s − i·48-s − 49-s − i·52-s − 2·61-s + 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (974, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.220709585\)
\(L(\frac12)\) \(\approx\) \(1.220709585\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.25459188500847120213976126290, −9.145070378795872887496922649413, −7.937506276045764260070505889456, −7.71768273400817361388107516167, −6.59237856878386894559391487066, −6.07656575759035098726143849334, −5.06959454151953004867126607954, −3.35201431561297263149031426010, −2.53427834112161155281039054535, −1.32628486048628880277635760772, 1.96214344693111957840067107755, 3.11187708242688814260298900724, 4.07246558130023193839670554749, 5.13072896459162593903227548049, 6.08149711375738883685380744832, 6.87859619786778195954961749199, 7.87464050505085899264970070378, 8.841380063270587955574540273338, 9.600924897993804394861135573876, 10.46020180672746986598322320198

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