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Properties

Label 2-975-13.9-c1-0-18
Degree $2$
Conductor $975$
Sign $0.859 - 0.511i$
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  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (0.500 + 0.866i)4-s + (0.499 + 0.866i)6-s + (1 + 1.73i)7-s + 3·8-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)11-s − 12-s + (3.5 + 0.866i)13-s + 1.99·14-s + (0.500 − 0.866i)16-s + (−3.5 − 6.06i)17-s − 0.999·18-s + (3 + 5.19i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (0.250 + 0.433i)4-s + (0.204 + 0.353i)6-s + (0.377 + 0.654i)7-s + 1.06·8-s + (−0.166 − 0.288i)9-s + (0.301 − 0.522i)11-s − 0.288·12-s + (0.970 + 0.240i)13-s + 0.534·14-s + (0.125 − 0.216i)16-s + (−0.848 − 1.47i)17-s − 0.235·18-s + (0.688 + 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06286 + 0.566944i\)
\(L(\frac12)\) \(\approx\) \(2.06286 + 0.566944i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 0.866i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.5 + 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.21818677558584196905902620188, −9.353207798472219718059114735473, −8.471658067851327598948094056383, −7.68070126782303781015585045849, −6.53331360353935987321803524227, −5.61977205413655949854890441339, −4.63377611233615817552777812590, −3.70705435397126112206404499671, −2.84936832497497622174317357224, −1.52046856734953241814898679373, 1.04866736705705732431020383137, 2.16939868031340990616177510771, 3.97849771921763688101968870004, 4.76306152758463431978843986820, 5.81267216774949321709633951043, 6.57136884557580550893840586768, 7.11476239730583052951184189369, 8.073787969366282799835236718343, 8.878815298447009765257447612629, 10.28747122814160851206407319985

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