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Properties

Label 2-975-195.158-c0-0-1
Degree $2$
Conductor $975$
Sign $-0.155 + 0.987i$
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  + (0.258 − 0.965i)3-s + (0.5 − 0.866i)4-s + (−0.258 + 0.448i)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 + (1.86 − 0.5i)19-s + (0.366 + 0.366i)21-s + (−0.707 + 0.707i)27-s + (0.258 + 0.448i)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  + (0.258 − 0.965i)3-s + (0.5 − 0.866i)4-s + (−0.258 + 0.448i)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 + (1.86 − 0.5i)19-s + (0.366 + 0.366i)21-s + (−0.707 + 0.707i)27-s + (0.258 + 0.448i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.155 + 0.987i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ -0.155 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.132268771\)
\(L(\frac12)\) \(\approx\) \(1.132268771\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
13 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 - 1.93iT - T^{2} \)
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

−9.824988888341178959864228019777, −9.285992316778043043908893602680, −8.192059285793527442836484143342, −7.27739480508224049421714302544, −6.73783525158588567517770063832, −5.64085023106563193840723572574, −5.16644815281216447131674865481, −3.21751600650039139480953801431, −2.42070901522517574952781215323, −1.11764943012564138788271240417, 2.25007561007341081224053506907, 3.34886063199728801307439432188, 4.00541901666332915746601791377, 5.06407548403679158825530665443, 6.16052231433560935275064977749, 7.39834305516403778198639243437, 7.74355284750463550902928893210, 8.964640445013579206085040621686, 9.565244410723031016431237163488, 10.34013256862564174739954578426

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