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Properties

Label 2-975-13.12-c1-0-31
Degree $2$
Conductor $975$
Sign $-0.277 + 0.960i$
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  − 1.73i·2-s + 3-s − 0.999·4-s − 1.73i·6-s + 3.46i·7-s − 1.73i·8-s + 9-s − 3.46i·11-s − 0.999·12-s + (1 − 3.46i)13-s + 5.99·14-s − 5·16-s + 6·17-s − 1.73i·18-s + 3.46i·19-s + ⋯
L(s)  = 1  − 1.22i·2-s + 0.577·3-s − 0.499·4-s − 0.707i·6-s + 1.30i·7-s − 0.612i·8-s + 0.333·9-s − 1.04i·11-s − 0.288·12-s + (0.277 − 0.960i)13-s + 1.60·14-s − 1.25·16-s + 1.45·17-s − 0.408i·18-s + 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.277 + 0.960i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28354 - 1.70647i\)
\(L(\frac12)\) \(\approx\) \(1.28354 - 1.70647i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
13 \( 1 + (-1 + 3.46i)T \)
good2 \( 1 + 1.73iT - 2T^{2} \)
7 \( 1 - 3.46iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−9.950883509439329918928548855049, −9.012525604542732739739282197235, −8.395421513078788980515219485775, −7.50631960058184002653580088982, −6.05462746589581197484607584344, −5.47194297151417843974516966826, −3.88265266079115475206211285074, −3.10809022980903272446505129450, −2.40868098729543463584390781909, −1.05953510937097008657874850035, 1.49808899212528247292178915742, 3.04262210108616053212904394019, 4.35179957296806594844870933078, 4.93266163837314642554371749871, 6.38538022154967726230096877373, 6.93548174906554347930325070034, 7.63570519375726352553717664062, 8.234756452490219312964017805000, 9.316193324389101243668353090925, 10.00467916272955544818941607342

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