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Properties

Label 2-975-195.62-c1-0-16
Degree $2$
Conductor $975$
Sign $0.847 + 0.530i$
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  + (0.442 − 1.65i)2-s + (−1.56 − 0.744i)3-s + (−0.796 − 0.459i)4-s + (−1.92 + 2.25i)6-s + (−0.0293 − 0.109i)7-s + (1.30 − 1.30i)8-s + (1.89 + 2.32i)9-s + (2.05 + 3.55i)11-s + (0.902 + 1.31i)12-s + (1.57 + 3.24i)13-s − 0.193·14-s + (−2.49 − 4.32i)16-s + (1.35 + 5.04i)17-s + (4.67 − 2.09i)18-s + (−3.97 + 6.88i)19-s + ⋯
L(s)  = 1  + (0.312 − 1.16i)2-s + (−0.902 − 0.429i)3-s + (−0.398 − 0.229i)4-s + (−0.783 + 0.919i)6-s + (−0.0110 − 0.0413i)7-s + (0.461 − 0.461i)8-s + (0.630 + 0.776i)9-s + (0.619 + 1.07i)11-s + (0.260 + 0.378i)12-s + (0.435 + 0.900i)13-s − 0.0517·14-s + (−0.624 − 1.08i)16-s + (0.327 + 1.22i)17-s + (1.10 − 0.493i)18-s + (−0.912 + 1.58i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.847 + 0.530i$
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: \((2,\ 975,\ (\ :1/2),\ 0.847 + 0.530i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40017 - 0.402401i\)
\(L(\frac12)\) \(\approx\) \(1.40017 - 0.402401i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.56 + 0.744i)T \)
5 \( 1 \)
13 \( 1 + (-1.57 - 3.24i)T \)
good2 \( 1 + (-0.442 + 1.65i)T + (-1.73 - i)T^{2} \)
7 \( 1 + (0.0293 + 0.109i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.05 - 3.55i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.35 - 5.04i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.97 - 6.88i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.79 - 6.69i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.00 - 3.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.89iT - 31T^{2} \)
37 \( 1 + (-4.10 - 1.09i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.73 + 3.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.30 + 4.88i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.185 - 0.185i)T + 47iT^{2} \)
53 \( 1 + (1.94 + 1.94i)T + 53iT^{2} \)
59 \( 1 + (-0.619 - 0.357i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.04 + 3.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.85 - 0.764i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.25 - 7.36i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.00 + 3.00i)T + 73iT^{2} \)
79 \( 1 - 4.84iT - 79T^{2} \)
83 \( 1 + (2.04 - 2.04i)T - 83iT^{2} \)
89 \( 1 + (2.79 - 1.61i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.0386 + 0.144i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.18278964393214512005953927713, −9.587922130133239064525035820449, −8.210308601942956847133581175672, −7.25644471290111043987204611020, −6.47107281670445023467110436898, −5.57402266625971129804718292514, −4.20047270226910466184813735379, −3.86836302242172819766866344654, −1.96958924275208033889799429271, −1.50865299065394649403632876854, 0.73310780446462516809445912461, 2.90504595049548058874077189697, 4.34466562216734154310092287656, 4.99786942258356398816800905304, 5.96649905150739481978849194506, 6.43724015669587526985403173419, 7.22909277113802286626085054269, 8.364858840844239383771710475725, 9.016303362881236570688216720277, 10.21278151558875441900016526596

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