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Properties

Label 2-975-65.29-c1-0-42
Degree $2$
Conductor $975$
Sign $-0.616 + 0.787i$
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  + (2.25 − 1.30i)2-s + (0.866 − 0.5i)3-s + (2.38 − 4.12i)4-s + (1.30 − 2.25i)6-s + (−3.11 − 1.80i)7-s − 7.20i·8-s + (0.499 − 0.866i)9-s + (2.60 + 4.50i)11-s − 4.76i·12-s + (−1.97 − 3.01i)13-s − 9.37·14-s + (−4.60 − 7.97i)16-s + (2.54 + 1.46i)17-s − 2.60i·18-s + (3.38 − 5.86i)19-s + ⋯
L(s)  = 1  + (1.59 − 0.919i)2-s + (0.499 − 0.288i)3-s + (1.19 − 2.06i)4-s + (0.531 − 0.919i)6-s + (−1.17 − 0.680i)7-s − 2.54i·8-s + (0.166 − 0.288i)9-s + (0.784 + 1.35i)11-s − 1.37i·12-s + (−0.547 − 0.837i)13-s − 2.50·14-s + (−1.15 − 1.99i)16-s + (0.616 + 0.356i)17-s − 0.613i·18-s + (0.776 − 1.34i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.616 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.616 + 0.787i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.616 + 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83678 - 3.76856i\)
\(L(\frac12)\) \(\approx\) \(1.83678 - 3.76856i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (1.97 + 3.01i)T \)
good2 \( 1 + (-2.25 + 1.30i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (3.11 + 1.80i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.60 - 4.50i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.54 - 1.46i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.38 + 5.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.79 - 2.76i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.916 - 1.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.10T + 31T^{2} \)
37 \( 1 + (3.06 - 1.76i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.68 - 4.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.74 - 1.58i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.80iT - 47T^{2} \)
53 \( 1 - 5.20iT - 53T^{2} \)
59 \( 1 + (3.68 - 6.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.71 + 2.97i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.03 + 1.75i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.85 + 8.40i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.805iT - 73T^{2} \)
79 \( 1 - 4.10T + 79T^{2} \)
83 \( 1 - 11.5iT - 83T^{2} \)
89 \( 1 + (-4.91 - 8.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.82 - 2.78i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.823050981953172597640457598494, −9.474799795173081155550581227128, −7.68468501104734676091881682125, −6.90169369931188687042648930534, −6.19101868167463437536542928715, −5.03249062074373526756857544278, −4.14823626077259985132257004513, −3.31956630627467295348186091315, −2.54882305049903936423742164354, −1.19515049846555443343075281532, 2.47028498583580453783136411206, 3.47100019271864323664673038270, 3.91066277518713944520574973879, 5.21460945637811184164494672203, 6.02933046195542130471378942839, 6.52048757405195690227759749191, 7.57409208402614989864638101647, 8.440372881191313918820723587286, 9.302529586352043463243127905050, 10.19553874403036396139339448024

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