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Properties

Label 2-975-195.167-c0-0-1
Degree $2$
Conductor $975$
Sign $-0.0320 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0320 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0320 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.0320 + 0.999i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ -0.0320 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8882996162\)
\(L(\frac12)\) \(\approx\) \(0.8882996162\)
\(L(1)\) 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

−10.15038757406011153761720966346, −9.604771180728428468212866752284, −7.957577604566815718112111915348, −7.27079679632430611871617541209, −6.70418147902140100516070189872, −5.67589306032189978982662107493, −4.79469797197852042278808615165, −4.08986563174561049671163148132, −2.01692626198572854955352423430, −1.02137752544723380666348283364, 1.98066638880397077597535979469, 3.02520461936076996854922964840, 4.43240964200390658385865804437, 5.32264856220692344149605842177, 5.94983018709411848850862996572, 7.09812233274576701276015421594, 7.84394170868152666664248234795, 8.778293058077922621474412170332, 9.494901505879654070588031572815, 10.81816982999094923778048672683

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