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Properties

Label 2-975-195.23-c1-0-74
Degree $2$
Conductor $975$
Sign $-0.722 + 0.691i$
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  + (1.85 + 0.496i)2-s + (−1.71 − 0.211i)3-s + (1.45 + 0.837i)4-s + (−3.07 − 1.24i)6-s + (−1.97 + 0.527i)7-s + (−0.440 − 0.440i)8-s + (2.91 + 0.727i)9-s + (0.0324 + 0.0561i)11-s + (−2.31 − 1.74i)12-s + (−1.22 − 3.39i)13-s − 3.91·14-s + (−2.27 − 3.93i)16-s + (−4.88 + 1.31i)17-s + (5.02 + 2.79i)18-s + (−1.37 + 2.38i)19-s + ⋯
L(s)  = 1  + (1.30 + 0.350i)2-s + (−0.992 − 0.122i)3-s + (0.725 + 0.418i)4-s + (−1.25 − 0.508i)6-s + (−0.744 + 0.199i)7-s + (−0.155 − 0.155i)8-s + (0.970 + 0.242i)9-s + (0.00977 + 0.0169i)11-s + (−0.668 − 0.504i)12-s + (−0.338 − 0.940i)13-s − 1.04·14-s + (−0.567 − 0.983i)16-s + (−1.18 + 0.317i)17-s + (1.18 + 0.657i)18-s + (−0.315 + 0.546i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.722 + 0.691i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192858 - 0.480107i\)
\(L(\frac12)\) \(\approx\) \(0.192858 - 0.480107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 + 0.211i)T \)
5 \( 1 \)
13 \( 1 + (1.22 + 3.39i)T \)
good2 \( 1 + (-1.85 - 0.496i)T + (1.73 + i)T^{2} \)
7 \( 1 + (1.97 - 0.527i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.0324 - 0.0561i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (4.88 - 1.31i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.37 - 2.38i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.10 + 1.36i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.40 + 2.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.312iT - 31T^{2} \)
37 \( 1 + (-0.326 + 1.21i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.66 + 9.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.44 + 2.53i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.638 - 0.638i)T - 47iT^{2} \)
53 \( 1 + (-5.43 + 5.43i)T - 53iT^{2} \)
59 \( 1 + (5.86 + 3.38i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.38 - 2.39i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.83 - 10.5i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (6.90 - 11.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.38 - 7.38i)T - 73iT^{2} \)
79 \( 1 + 5.34iT - 79T^{2} \)
83 \( 1 + (-2.60 - 2.60i)T + 83iT^{2} \)
89 \( 1 + (-12.5 + 7.22i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.12 + 1.90i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.982699509729196267756200914349, −8.880267474880912239461283625914, −7.58677051143108600509171384884, −6.73674319571150088732258595361, −6.00282726731911796860723742332, −5.52080861143514052578448985493, −4.45120628848659202219648706126, −3.72735893307431284835463344037, −2.34435930535565011284324161678, −0.16176822103168107957820001800, 2.00799764970255805173427521503, 3.31589036351226721716801868373, 4.40608744850535854942566855241, 4.76537155226176946407985958178, 6.05347829398218110386836312989, 6.44028206365384755574760381460, 7.38650466113685373104163612090, 8.917652676122035919540546218347, 9.651117629741572331028059810527, 10.67257683259101790808465558323

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