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Properties

Label 2-975-13.10-c1-0-16
Degree $2$
Conductor $975$
Sign $0.324 - 0.945i$
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  + (1.14 − 0.663i)2-s + (0.5 + 0.866i)3-s + (−0.119 + 0.207i)4-s + (1.14 + 0.663i)6-s + (−0.917 − 0.529i)7-s + 2.97i·8-s + (−0.499 + 0.866i)9-s + (2.15 − 1.24i)11-s − 0.239·12-s + (−2.68 + 2.40i)13-s − 1.40·14-s + (1.73 + 3.00i)16-s + (−2.47 + 4.28i)17-s + 1.32i·18-s + (5.50 + 3.17i)19-s + ⋯
L(s)  = 1  + (0.812 − 0.469i)2-s + (0.288 + 0.499i)3-s + (−0.0598 + 0.103i)4-s + (0.469 + 0.270i)6-s + (−0.346 − 0.200i)7-s + 1.05i·8-s + (−0.166 + 0.288i)9-s + (0.650 − 0.375i)11-s − 0.0690·12-s + (−0.745 + 0.666i)13-s − 0.375·14-s + (0.433 + 0.750i)16-s + (−0.600 + 1.04i)17-s + 0.312i·18-s + (1.26 + 0.729i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.324 - 0.945i$
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: \((2,\ 975,\ (\ :1/2),\ 0.324 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80523 + 1.28898i\)
\(L(\frac12)\) \(\approx\) \(1.80523 + 1.28898i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.68 - 2.40i)T \)
good2 \( 1 + (-1.14 + 0.663i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (0.917 + 0.529i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.15 + 1.24i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.47 - 4.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.50 - 3.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.81 - 3.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.117 - 0.202i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.31iT - 31T^{2} \)
37 \( 1 + (-5.29 + 3.05i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.66 - 4.42i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.83 - 8.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.70iT - 47T^{2} \)
53 \( 1 - 4.98T + 53T^{2} \)
59 \( 1 + (-1.82 - 1.05i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.52 - 2.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.61 + 3.81i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.08 - 4.08i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.98iT - 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 + 15.8iT - 83T^{2} \)
89 \( 1 + (-13.7 + 7.92i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.70 + 5.02i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.12398532683760190376168052006, −9.456143765294251891117163123355, −8.586339283876017807792584933393, −7.77159652323816408617299755829, −6.65200168677443706682799978916, −5.58178854535134755783113054809, −4.66357128103915404161555093793, −3.80164373142329773161357272594, −3.18141407871407084934227226338, −1.86499684595656449246613168596, 0.78031352461921110228190234037, 2.53915471749361913494725909081, 3.54779789269564275168351980319, 4.81156034974401370439333522104, 5.35113912873387993598215336173, 6.64328145730955705352516196972, 6.92351867047122778815109978150, 7.938569342557749502240029530655, 9.257162610732236097751917047323, 9.525561144646082634558897730015

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