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Properties

Label 2-975-195.17-c1-0-16
Degree $2$
Conductor $975$
Sign $-0.853 - 0.520i$
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  + (−1.65 + 0.442i)2-s + (0.744 + 1.56i)3-s + (0.796 − 0.459i)4-s + (−1.92 − 2.25i)6-s + (−0.109 − 0.0293i)7-s + (1.30 − 1.30i)8-s + (−1.89 + 2.32i)9-s + (2.05 − 3.55i)11-s + (1.31 + 0.902i)12-s + (−3.24 − 1.57i)13-s + 0.193·14-s + (−2.49 + 4.32i)16-s + (5.04 + 1.35i)17-s + (2.09 − 4.67i)18-s + (3.97 + 6.88i)19-s + ⋯
L(s)  = 1  + (−1.16 + 0.312i)2-s + (0.429 + 0.902i)3-s + (0.398 − 0.229i)4-s + (−0.783 − 0.919i)6-s + (−0.0413 − 0.0110i)7-s + (0.461 − 0.461i)8-s + (−0.630 + 0.776i)9-s + (0.619 − 1.07i)11-s + (0.378 + 0.260i)12-s + (−0.900 − 0.435i)13-s + 0.0517·14-s + (−0.624 + 1.08i)16-s + (1.22 + 0.327i)17-s + (0.493 − 1.10i)18-s + (0.912 + 1.58i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.853 - 0.520i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.853 - 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.195382 + 0.695880i\)
\(L(\frac12)\) \(\approx\) \(0.195382 + 0.695880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.744 - 1.56i)T \)
5 \( 1 \)
13 \( 1 + (3.24 + 1.57i)T \)
good2 \( 1 + (1.65 - 0.442i)T + (1.73 - i)T^{2} \)
7 \( 1 + (0.109 + 0.0293i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.05 + 3.55i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-5.04 - 1.35i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.97 - 6.88i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.69 - 1.79i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.00 - 3.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.89iT - 31T^{2} \)
37 \( 1 + (-1.09 - 4.10i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.73 - 3.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.88 - 1.30i)T + (37.2 + 21.5i)T^{2} \)
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 \( 1 + (-0.764 - 2.85i)T + (-58.0 + 33.5i)T^{2} \)
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 \( 1 + (0.144 + 0.0386i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.944361334192884592621430646902, −9.691487510442718912168978597615, −8.641306743327460739392704157950, −8.068295654688486528873968935711, −7.47111248812367399469282792319, −6.08582536329852923927008976838, −5.24934066388070520463722847466, −3.89548673453751074256618369953, −3.22562910044153592463397397067, −1.40613104047697674050585987635, 0.50234094667453441025366082788, 1.83394977657436012435952797570, 2.62848603352889275446695120910, 4.21182900394315731339384209409, 5.40179586296891542916898249248, 6.68621975203574060491188758774, 7.54146331096791699612953465835, 7.82304774197597465565491240617, 9.096722847126628861253125125946, 9.492031306630090850628237530584

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