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Properties

Label 2-975-975.536-c0-0-1
Degree $2$
Conductor $975$
Sign $0.943 + 0.332i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.251 − 0.564i)2-s + (−0.207 − 0.978i)3-s + (0.413 + 0.459i)4-s + (0.587 + 0.809i)5-s + (−0.604 − 0.128i)6-s + (−0.5 + 0.866i)7-s + (0.951 − 0.309i)8-s + (−0.913 + 0.406i)9-s + (0.604 − 0.128i)10-s + (−0.251 + 0.564i)11-s + (0.363 − 0.500i)12-s + (0.809 + 0.587i)13-s + (0.363 + 0.5i)14-s + (0.669 − 0.743i)15-s + (0.207 − 0.978i)17-s + 0.618i·18-s + ⋯
L(s)  = 1  + (0.251 − 0.564i)2-s + (−0.207 − 0.978i)3-s + (0.413 + 0.459i)4-s + (0.587 + 0.809i)5-s + (−0.604 − 0.128i)6-s + (−0.5 + 0.866i)7-s + (0.951 − 0.309i)8-s + (−0.913 + 0.406i)9-s + (0.604 − 0.128i)10-s + (−0.251 + 0.564i)11-s + (0.363 − 0.500i)12-s + (0.809 + 0.587i)13-s + (0.363 + 0.5i)14-s + (0.669 − 0.743i)15-s + (0.207 − 0.978i)17-s + 0.618i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.943 + 0.332i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (536, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.943 + 0.332i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.279857563\)
\(L(\frac12)\) \(\approx\) \(1.279857563\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.207 + 0.978i)T \)
5 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (-0.251 + 0.564i)T + (-0.669 - 0.743i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.251 - 0.564i)T + (-0.669 - 0.743i)T^{2} \)
17 \( 1 + (-0.207 + 0.978i)T + (-0.913 - 0.406i)T^{2} \)
19 \( 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2} \)
23 \( 1 + (-0.406 + 0.913i)T + (-0.669 - 0.743i)T^{2} \)
29 \( 1 + (-0.913 + 0.406i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.104 + 0.994i)T + (-0.978 - 0.207i)T^{2} \)
41 \( 1 + (0.994 + 0.104i)T + (0.978 + 0.207i)T^{2} \)
43 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.406 + 0.913i)T + (-0.669 + 0.743i)T^{2} \)
61 \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \)
67 \( 1 + (1.08 - 1.20i)T + (-0.104 - 0.994i)T^{2} \)
71 \( 1 + (0.104 - 0.994i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.406 - 0.913i)T + (-0.669 - 0.743i)T^{2} \)
97 \( 1 + (-1.08 - 1.20i)T + (-0.104 + 0.994i)T^{2} \)
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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

−10.49532534607648306377357625250, −9.350634368551580223452092618959, −8.480620859979538329205925886516, −7.40063030061922102784655756719, −6.70161394718525419257332918664, −6.18955800663698371176257892466, −4.99065205295133210476015724103, −3.45447644636570410380277797726, −2.49866972560658197034910850023, −1.94892593838254745935304213732, 1.34833703054177460407003292661, 3.19842615994455229981655452242, 4.27455009460801905057203280900, 5.12545407904031986609511317483, 6.05885634015204491970657165501, 6.33664424134992453016813456513, 7.84535141305924591196023054715, 8.566155385025508983737429915814, 9.596296458182531807710934648558, 10.34646064029638195411847719535

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