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Properties

Label 2-975-195.38-c1-0-2
Degree $2$
Conductor $975$
Sign $-0.960 - 0.277i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.25 + 1.25i)2-s + (1.06 − 1.36i)3-s − 1.15i·4-s + (0.381 + 3.05i)6-s + (−1.60 − 1.60i)7-s + (−1.05 − 1.05i)8-s + (−0.738 − 2.90i)9-s − 2.48·11-s + (−1.58 − 1.23i)12-s + (−0.624 + 3.55i)13-s + 4.02·14-s + 4.97·16-s + (1.27 + 1.27i)17-s + (4.58 + 2.72i)18-s + 1.93·19-s + ⋯
L(s)  = 1  + (−0.888 + 0.888i)2-s + (0.613 − 0.789i)3-s − 0.579i·4-s + (0.155 + 1.24i)6-s + (−0.605 − 0.605i)7-s + (−0.373 − 0.373i)8-s + (−0.246 − 0.969i)9-s − 0.748·11-s + (−0.457 − 0.355i)12-s + (−0.173 + 0.984i)13-s + 1.07·14-s + 1.24·16-s + (0.309 + 0.309i)17-s + (1.08 + 0.642i)18-s + 0.443·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.960 - 0.277i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (818, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.960 - 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0413133 + 0.291456i\)
\(L(\frac12)\) \(\approx\) \(0.0413133 + 0.291456i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.06 + 1.36i)T \)
5 \( 1 \)
13 \( 1 + (0.624 - 3.55i)T \)
good2 \( 1 + (1.25 - 1.25i)T - 2iT^{2} \)
7 \( 1 + (1.60 + 1.60i)T + 7iT^{2} \)
11 \( 1 + 2.48T + 11T^{2} \)
17 \( 1 + (-1.27 - 1.27i)T + 17iT^{2} \)
19 \( 1 - 1.93T + 19T^{2} \)
23 \( 1 + (5.56 - 5.56i)T - 23iT^{2} \)
29 \( 1 + 9.66T + 29T^{2} \)
31 \( 1 - 6.61iT - 31T^{2} \)
37 \( 1 + (-3.53 - 3.53i)T + 37iT^{2} \)
41 \( 1 - 7.35T + 41T^{2} \)
43 \( 1 + (-3.46 - 3.46i)T + 43iT^{2} \)
47 \( 1 + (4.55 - 4.55i)T - 47iT^{2} \)
53 \( 1 + (3.97 - 3.97i)T - 53iT^{2} \)
59 \( 1 - 2.79iT - 59T^{2} \)
61 \( 1 + 2.54T + 61T^{2} \)
67 \( 1 + (1.34 + 1.34i)T + 67iT^{2} \)
71 \( 1 + 6.58T + 71T^{2} \)
73 \( 1 + (-2.82 + 2.82i)T - 73iT^{2} \)
79 \( 1 - 6.48iT - 79T^{2} \)
83 \( 1 + (-3.17 - 3.17i)T + 83iT^{2} \)
89 \( 1 + 7.47iT - 89T^{2} \)
97 \( 1 + (9.27 + 9.27i)T + 97iT^{2} \)
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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

−9.755267417142640885638714560506, −9.528353430820723325273536269855, −8.552858780310934844310825613597, −7.60820195491472939458927146581, −7.41994941436599598412400973541, −6.44948523757964435867371978689, −5.71333891353893561985636157964, −3.97590179327058235844161674614, −3.03938440693348914067603044916, −1.49444729401967876410334665141, 0.16389591970516038518976631067, 2.24602124782070832764725456518, 2.82318370510624019043212219860, 3.87225011594347989953313853466, 5.31548698811857014486214918766, 5.91364971811874196700541047310, 7.73524278470630851516203525694, 8.107287523078234370733049046110, 9.285752601024123225362558836624, 9.502127132796374542156665677963

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