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Properties

Label 2-975-13.10-c1-0-0
Degree $2$
Conductor $975$
Sign $-0.831 - 0.555i$
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  + (1.27 − 0.733i)2-s + (0.5 + 0.866i)3-s + (0.0756 − 0.131i)4-s + (1.27 + 0.733i)6-s + (−3.75 − 2.16i)7-s + 2.71i·8-s + (−0.499 + 0.866i)9-s + (−5.05 + 2.91i)11-s + 0.151·12-s + (−1.63 − 3.21i)13-s − 6.35·14-s + (2.13 + 3.70i)16-s + (−1.43 + 2.49i)17-s + 1.46i·18-s + (−2.89 − 1.66i)19-s + ⋯
L(s)  = 1  + (0.898 − 0.518i)2-s + (0.288 + 0.499i)3-s + (0.0378 − 0.0655i)4-s + (0.518 + 0.299i)6-s + (−1.41 − 0.818i)7-s + 0.958i·8-s + (−0.166 + 0.288i)9-s + (−1.52 + 0.879i)11-s + 0.0436·12-s + (−0.454 − 0.890i)13-s − 1.69·14-s + (0.534 + 0.926i)16-s + (−0.349 + 0.604i)17-s + 0.345i·18-s + (−0.663 − 0.382i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.831 - 0.555i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.831 - 0.555i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.180664 + 0.596282i\)
\(L(\frac12)\) \(\approx\) \(0.180664 + 0.596282i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (1.63 + 3.21i)T \)
good2 \( 1 + (-1.27 + 0.733i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (3.75 + 2.16i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.05 - 2.91i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.43 - 2.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.89 + 1.66i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.623 - 1.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.86 - 4.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.880iT - 31T^{2} \)
37 \( 1 + (-0.166 + 0.0960i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.198 + 0.114i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.47 + 4.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.0904iT - 47T^{2} \)
53 \( 1 - 4.46T + 53T^{2} \)
59 \( 1 + (6.48 + 3.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.78 - 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.46 + 3.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.08 - 1.77i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.62iT - 73T^{2} \)
79 \( 1 + 6.75T + 79T^{2} \)
83 \( 1 + 9.36iT - 83T^{2} \)
89 \( 1 + (13.1 - 7.60i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.32 + 0.762i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.45545974944617259776595089932, −9.831157131299611031389262298560, −8.739074121935382772631406939362, −7.80507887036723533328663017130, −6.98841557387696889563187893006, −5.72761778501724570698095659254, −4.85411405681006364629620124352, −4.04483478369670870486960894455, −3.11424233498467769471930188540, −2.44678745088432470793856640153, 0.18653020711764057132672894238, 2.46524546400851884221366794786, 3.20522236884163312335212504407, 4.46282447539961822495390219041, 5.51021277277438562422208942934, 6.20558842916197151909389257645, 6.78770108005105081920927244940, 7.82524710771854765401530445967, 8.846111275610823906366697647995, 9.583401895137648009237029274652

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