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Properties

Label 2-975-195.32-c0-0-1
Degree $2$
Conductor $975$
Sign $-0.418 + 0.908i$
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 + (1.67 − 0.965i)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 + (−1.67 − 0.965i)28-s + (−1 + i)31-s + (0.866 + 0.5i)36-s + (−1.22 − 0.707i)37-s + (−0.866 − 0.500i)39-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)3-s + (−0.5 − 0.866i)4-s + (1.67 − 0.965i)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 + (−1.67 − 0.965i)28-s + (−1 + i)31-s + (0.866 + 0.5i)36-s + (−1.22 − 0.707i)37-s + (−0.866 − 0.500i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.418 + 0.908i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ -0.418 + 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9496170772\)
\(L(\frac12)\) \(\approx\) \(0.9496170772\)
\(L(1)\) 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 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 - 0.517T + T^{2} \)
79 \( 1 - 1.73iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.24629499248698166755794004083, −8.863197411199286017042520815699, −8.209887458618786097117935413732, −7.49969104059670907437500920583, −6.54679839692868576526397170910, −5.49274259663099978583306122785, −4.96272480173890854125346326873, −3.76635340517181345846540417150, −1.86390255600461271548844928621, −1.08166705411141212548518626572, 2.10059194820829656696200078256, 3.45128989055165716164754111912, 4.42137284416584587507306714259, 5.02518205011038637528395133447, 5.91016989799203008826724077643, 7.28038594460905759569777442576, 8.372463039698783708113353584018, 8.731228507463052723745123665049, 9.390298032171206689928982463980, 10.59584132741676083357994678129

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