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Properties

Label 2-975-195.62-c1-0-30
Degree $2$
Conductor $975$
Sign $-0.318 - 0.947i$
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  + (−0.0288 + 0.107i)2-s + (−1.51 + 0.836i)3-s + (1.72 + 0.993i)4-s + (−0.0462 − 0.187i)6-s + (1.03 + 3.86i)7-s + (−0.314 + 0.314i)8-s + (1.60 − 2.53i)9-s + (2.60 + 4.51i)11-s + (−3.44 − 0.0681i)12-s + (3.33 − 1.36i)13-s − 0.445·14-s + (1.96 + 3.39i)16-s + (−0.878 − 3.27i)17-s + (0.226 + 0.245i)18-s + (0.210 − 0.364i)19-s + ⋯
L(s)  = 1  + (−0.0203 + 0.0760i)2-s + (−0.875 + 0.482i)3-s + (0.860 + 0.496i)4-s + (−0.0188 − 0.0764i)6-s + (0.391 + 1.45i)7-s + (−0.111 + 0.111i)8-s + (0.533 − 0.845i)9-s + (0.785 + 1.36i)11-s + (−0.993 − 0.0196i)12-s + (0.926 − 0.377i)13-s − 0.118·14-s + (0.490 + 0.849i)16-s + (−0.212 − 0.794i)17-s + (0.0534 + 0.0578i)18-s + (0.0482 − 0.0836i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.318 - 0.947i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.318 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.942618 + 1.31154i\)
\(L(\frac12)\) \(\approx\) \(0.942618 + 1.31154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.51 - 0.836i)T \)
5 \( 1 \)
13 \( 1 + (-3.33 + 1.36i)T \)
good2 \( 1 + (0.0288 - 0.107i)T + (-1.73 - i)T^{2} \)
7 \( 1 + (-1.03 - 3.86i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.60 - 4.51i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.878 + 3.27i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.210 + 0.364i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.705 + 2.63i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.52 + 4.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.23iT - 31T^{2} \)
37 \( 1 + (-5.74 - 1.53i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.973 + 1.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.242 - 0.905i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.29 - 1.29i)T + 47iT^{2} \)
53 \( 1 + (2.45 + 2.45i)T + 53iT^{2} \)
59 \( 1 + (7.83 + 4.52i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.65 + 4.60i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.41 + 1.98i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.09 - 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.29 + 1.29i)T + 73iT^{2} \)
79 \( 1 + 2.49iT - 79T^{2} \)
83 \( 1 + (9.55 - 9.55i)T - 83iT^{2} \)
89 \( 1 + (-3.74 + 2.16i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.740 - 2.76i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.35199379277020647329059206969, −9.408825232105557494613065958693, −8.718554415960152482214219783810, −7.63614710460232138993536187035, −6.66286004158569528679697498389, −6.06548399592476734287186296342, −5.12051792381710038467023831846, −4.15342897916198468012566499451, −2.86012543341027896823911910943, −1.67885687860287618375571166363, 0.929026049716043861199786981329, 1.61898603573928269272862430907, 3.46411056175834536077492947628, 4.43024930791644425775779771831, 5.84923667985080640475464052176, 6.20263464871626456025113885272, 7.12745677739190735867539238366, 7.77319884655703273832814513463, 8.909455400341128564014211893897, 10.16895771654996088728011733663

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