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Properties

Label 2-975-195.188-c0-0-1
Degree $2$
Conductor $975$
Sign $0.985 + 0.168i$
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  + (0.965 − 0.258i)3-s + (0.5 + 0.866i)4-s + (−0.965 − 1.67i)7-s + (0.866 − 0.499i)9-s + (0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s + (−0.499 + 0.866i)16-s + (0.133 − 0.5i)19-s + (−1.36 − 1.36i)21-s + (0.707 − 0.707i)27-s + (0.965 − 1.67i)28-s + (−1 + i)31-s + (0.866 + 0.5i)36-s + (−0.707 + 1.22i)37-s + (0.866 + 0.500i)39-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)3-s + (0.5 + 0.866i)4-s + (−0.965 − 1.67i)7-s + (0.866 − 0.499i)9-s + (0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s + (−0.499 + 0.866i)16-s + (0.133 − 0.5i)19-s + (−1.36 − 1.36i)21-s + (0.707 − 0.707i)27-s + (0.965 − 1.67i)28-s + (−1 + i)31-s + (0.866 + 0.5i)36-s + (−0.707 + 1.22i)37-s + (0.866 + 0.500i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.985 + 0.168i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (968, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.985 + 0.168i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.434448706\)
\(L(\frac12)\) \(\approx\) \(1.434448706\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
13 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (1 - i)T - iT^{2} \)
37 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 - 0.517iT - T^{2} \)
79 \( 1 + 1.73iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
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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

−10.17592662103846104969600805331, −9.226926769867931777274210260199, −8.477760962946037559038856712189, −7.54703844129560173548025069099, −6.92652754370250159918298733982, −6.48602170712149629944000027071, −4.47620457707915903865424607222, −3.60948659875300300186220765889, −3.12121027568955972087689654136, −1.59413336443750033901481329738, 1.88352875578161830734040203567, 2.76667054971935273830991532849, 3.68084214617494216929492109472, 5.27489382363038456803309941301, 5.85427475642332907204113343324, 6.74427996403591083974713342352, 7.86383151283660690967436765958, 8.803737681292537267556431372934, 9.372599684032123122977317022663, 10.03837408367806827161558803339

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