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Properties

Label 2-975-15.2-c1-0-22
Degree $2$
Conductor $975$
Sign $0.969 + 0.245i$
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  + (1.51 − 1.51i)2-s + (−1.72 + 0.145i)3-s − 2.61i·4-s + (−2.39 + 2.84i)6-s + (0.988 + 0.988i)7-s + (−0.928 − 0.928i)8-s + (2.95 − 0.503i)9-s + 5.57i·11-s + (0.381 + 4.50i)12-s + (−0.707 + 0.707i)13-s + 3.00·14-s + 2.40·16-s + (−3.09 + 3.09i)17-s + (3.72 − 5.25i)18-s + 1.37i·19-s + ⋯
L(s)  = 1  + (1.07 − 1.07i)2-s + (−0.996 + 0.0842i)3-s − 1.30i·4-s + (−0.979 + 1.16i)6-s + (0.373 + 0.373i)7-s + (−0.328 − 0.328i)8-s + (0.985 − 0.167i)9-s + 1.68i·11-s + (0.110 + 1.30i)12-s + (−0.196 + 0.196i)13-s + 0.802·14-s + 0.600·16-s + (−0.750 + 0.750i)17-s + (0.878 − 1.23i)18-s + 0.314i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.969 + 0.245i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.969 + 0.245i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05742 - 0.256551i\)
\(L(\frac12)\) \(\approx\) \(2.05742 - 0.256551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.145i)T \)
5 \( 1 \)
13 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (-1.51 + 1.51i)T - 2iT^{2} \)
7 \( 1 + (-0.988 - 0.988i)T + 7iT^{2} \)
11 \( 1 - 5.57iT - 11T^{2} \)
17 \( 1 + (3.09 - 3.09i)T - 17iT^{2} \)
19 \( 1 - 1.37iT - 19T^{2} \)
23 \( 1 + (-0.821 - 0.821i)T + 23iT^{2} \)
29 \( 1 - 1.96T + 29T^{2} \)
31 \( 1 - 7.33T + 31T^{2} \)
37 \( 1 + (-0.328 - 0.328i)T + 37iT^{2} \)
41 \( 1 - 6.15iT - 41T^{2} \)
43 \( 1 + (-6.64 + 6.64i)T - 43iT^{2} \)
47 \( 1 + (0.596 - 0.596i)T - 47iT^{2} \)
53 \( 1 + (8.25 + 8.25i)T + 53iT^{2} \)
59 \( 1 + 1.25T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + (5.90 + 5.90i)T + 67iT^{2} \)
71 \( 1 - 4.01iT - 71T^{2} \)
73 \( 1 + (9.99 - 9.99i)T - 73iT^{2} \)
79 \( 1 - 15.0iT - 79T^{2} \)
83 \( 1 + (-5.18 - 5.18i)T + 83iT^{2} \)
89 \( 1 + 5.14T + 89T^{2} \)
97 \( 1 + (7.83 + 7.83i)T + 97iT^{2} \)
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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

−10.13593071970704099700623780487, −9.772717314268093359829616203722, −8.323733163099486201717936773651, −7.16284478311788272611592640940, −6.27429624893146143706424894467, −5.24182114338490269723868960371, −4.60877514475351348174849824208, −3.97463606233657016675888203918, −2.40772982234420359254523843233, −1.52820634917684193694146326228, 0.837644194979953702228602636333, 3.02609520524422140529967459152, 4.32247426185099720177964485532, 4.86952609646356086337109751076, 5.85840969874961054423771522124, 6.32905422471284550522704441838, 7.23308693854905359483722369448, 7.949673830164220628510097320736, 9.011320526912711010159663556691, 10.31249064088068482454180723694

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