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Properties

Label 2-975-195.23-c1-0-23
Degree $2$
Conductor $975$
Sign $0.544 + 0.838i$
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  + (−0.254 − 0.0683i)2-s + (−1.56 − 0.750i)3-s + (−1.67 − 0.965i)4-s + (0.346 + 0.297i)6-s + (−3.34 + 0.894i)7-s + (0.733 + 0.733i)8-s + (1.87 + 2.34i)9-s + (0.378 + 0.655i)11-s + (1.88 + 2.76i)12-s + (−3.19 + 1.67i)13-s + 0.912·14-s + (1.79 + 3.10i)16-s + (7.08 − 1.89i)17-s + (−0.318 − 0.725i)18-s + (−2.37 + 4.11i)19-s + ⋯
L(s)  = 1  + (−0.180 − 0.0483i)2-s + (−0.901 − 0.433i)3-s + (−0.835 − 0.482i)4-s + (0.141 + 0.121i)6-s + (−1.26 + 0.338i)7-s + (0.259 + 0.259i)8-s + (0.624 + 0.780i)9-s + (0.114 + 0.197i)11-s + (0.544 + 0.796i)12-s + (−0.885 + 0.464i)13-s + 0.243·14-s + (0.448 + 0.776i)16-s + (1.71 − 0.460i)17-s + (−0.0749 − 0.170i)18-s + (−0.545 + 0.945i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.544 + 0.838i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.544 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.464374 - 0.252104i\)
\(L(\frac12)\) \(\approx\) \(0.464374 - 0.252104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.56 + 0.750i)T \)
5 \( 1 \)
13 \( 1 + (3.19 - 1.67i)T \)
good2 \( 1 + (0.254 + 0.0683i)T + (1.73 + i)T^{2} \)
7 \( 1 + (3.34 - 0.894i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.378 - 0.655i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-7.08 + 1.89i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.37 - 4.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.15 + 0.309i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.57 + 6.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.75iT - 31T^{2} \)
37 \( 1 + (-0.717 + 2.67i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.69 - 4.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.52 - 1.48i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.28 + 2.28i)T - 47iT^{2} \)
53 \( 1 + (-5.30 + 5.30i)T - 53iT^{2} \)
59 \( 1 + (-4.58 - 2.64i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.28 - 2.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.50 + 5.60i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.328 - 0.568i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.39 + 8.39i)T - 73iT^{2} \)
79 \( 1 + 6.35iT - 79T^{2} \)
83 \( 1 + (-2.33 - 2.33i)T + 83iT^{2} \)
89 \( 1 + (-9.05 + 5.22i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.55 - 2.29i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.817846200775140066988917469956, −9.458512063723945825444674337468, −8.111989166191414569300442848741, −7.33536790496058456738263723449, −6.19780096423871404789323310110, −5.73326644159615460625945284984, −4.75740585516507098905785620215, −3.67829584119664945142759408936, −2.05500886949686546945626709857, −0.50460841779208065942549337155, 0.72161545611503178981151405454, 3.17801636542973808709572931068, 3.83440170476846101690030341190, 4.96329735705777363170357258911, 5.69415427901705987211980437475, 6.80956639166065463072508183846, 7.48207096418013335132575196526, 8.658774763680390422261763880663, 9.531776148840800747921770109378, 10.04145483343014139067858740371

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