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Properties

Label 2-975-13.3-c1-0-42
Degree $2$
Conductor $975$
Sign $-0.767 - 0.641i$
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  + (−0.115 − 0.200i)2-s + (−0.5 − 0.866i)3-s + (0.973 − 1.68i)4-s + (−0.115 + 0.200i)6-s + (−0.580 + 1.00i)7-s − 0.914·8-s + (−0.499 + 0.866i)9-s + (−1.76 − 3.05i)11-s − 1.94·12-s + (−3.59 + 0.297i)13-s + 0.269·14-s + (−1.84 − 3.18i)16-s + (−3.08 + 5.34i)17-s + 0.231·18-s + (−3.63 + 6.29i)19-s + ⋯
L(s)  = 1  + (−0.0819 − 0.141i)2-s + (−0.288 − 0.499i)3-s + (0.486 − 0.842i)4-s + (−0.0472 + 0.0819i)6-s + (−0.219 + 0.380i)7-s − 0.323·8-s + (−0.166 + 0.288i)9-s + (−0.531 − 0.920i)11-s − 0.561·12-s + (−0.996 + 0.0824i)13-s + 0.0719·14-s + (−0.460 − 0.796i)16-s + (−0.748 + 1.29i)17-s + 0.0546·18-s + (−0.834 + 1.44i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.767 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.767 - 0.641i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.767 - 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0811961 + 0.223679i\)
\(L(\frac12)\) \(\approx\) \(0.0811961 + 0.223679i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.59 - 0.297i)T \)
good2 \( 1 + (0.115 + 0.200i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.580 - 1.00i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.76 + 3.05i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.08 - 5.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.63 - 6.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.180 + 0.313i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.95 + 8.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.83T + 31T^{2} \)
37 \( 1 + (1.85 + 3.21i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.08 + 5.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.59 + 2.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 - 3.46T + 53T^{2} \)
59 \( 1 + (2.74 - 4.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.361 + 0.626i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.56 - 4.44i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.237 + 0.411i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.75T + 73T^{2} \)
79 \( 1 + 4.59T + 79T^{2} \)
83 \( 1 + 8.41T + 83T^{2} \)
89 \( 1 + (-7.70 - 13.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.662 + 1.14i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.732627592420544845089197483767, −8.563015356672765163474350984088, −7.889003113867455062050755060551, −6.71483838765953644424125595062, −6.02457648945520579849916325234, −5.49918167453391357181823179994, −4.15133550425370072344721440607, −2.62392310482534199121332457204, −1.79194096687364081388391574887, −0.10436225394837667097878517923, 2.34447772205472033280282092731, 3.16958158876011045177931077731, 4.56862309270662781726831918029, 4.98676881525571720357368501420, 6.69876028223341360490224115767, 6.95154868737803947850484604635, 7.909870522009544868583968942744, 8.926889515071851260295911098726, 9.681042844586958191887890702762, 10.50218156131832270276822903586

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