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Properties

Label 2-975-975.386-c0-0-0
Degree $2$
Conductor $975$
Sign $0.344 - 0.938i$
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  + (−0.614 + 0.0646i)2-s + (−0.743 + 0.669i)3-s + (−0.604 + 0.128i)4-s + (0.587 + 0.809i)5-s + (0.413 − 0.459i)6-s + (−0.5 − 0.866i)7-s + (0.951 − 0.309i)8-s + (0.104 − 0.994i)9-s + (−0.413 − 0.459i)10-s + (0.614 − 0.0646i)11-s + (0.363 − 0.500i)12-s + (0.809 + 0.587i)13-s + (0.363 + 0.5i)14-s + (−0.978 − 0.207i)15-s + (0.743 + 0.669i)17-s + 0.618i·18-s + ⋯
L(s)  = 1  + (−0.614 + 0.0646i)2-s + (−0.743 + 0.669i)3-s + (−0.604 + 0.128i)4-s + (0.587 + 0.809i)5-s + (0.413 − 0.459i)6-s + (−0.5 − 0.866i)7-s + (0.951 − 0.309i)8-s + (0.104 − 0.994i)9-s + (−0.413 − 0.459i)10-s + (0.614 − 0.0646i)11-s + (0.363 − 0.500i)12-s + (0.809 + 0.587i)13-s + (0.363 + 0.5i)14-s + (−0.978 − 0.207i)15-s + (0.743 + 0.669i)17-s + 0.618i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.344 - 0.938i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (386, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.344 - 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5586751148\)
\(L(\frac12)\) \(\approx\) \(0.5586751148\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.743 - 0.669i)T \)
5 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (0.614 - 0.0646i)T + (0.978 - 0.207i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.614 + 0.0646i)T + (0.978 - 0.207i)T^{2} \)
17 \( 1 + (-0.743 - 0.669i)T + (0.104 + 0.994i)T^{2} \)
19 \( 1 + (-0.669 + 0.743i)T + (-0.104 - 0.994i)T^{2} \)
23 \( 1 + (0.994 - 0.104i)T + (0.978 - 0.207i)T^{2} \)
29 \( 1 + (0.104 - 0.994i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \)
41 \( 1 + (-0.406 - 0.913i)T + (-0.669 + 0.743i)T^{2} \)
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 \( 1 + (-0.994 - 0.104i)T + (0.978 + 0.207i)T^{2} \)
61 \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \)
67 \( 1 + (-1.58 - 0.336i)T + (0.913 + 0.406i)T^{2} \)
71 \( 1 + (-0.913 + 0.406i)T^{2} \)
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 \( 1 + (-0.994 + 0.104i)T + (0.978 - 0.207i)T^{2} \)
97 \( 1 + (1.58 - 0.336i)T + (0.913 - 0.406i)T^{2} \)
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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

−10.02052793627366882343862613536, −9.847955452604278775405622747736, −9.016853464386650908568469839006, −7.913895519012002177348151837756, −6.80892707630856917764921761737, −6.30717340994338275213959852508, −5.17869478898668723234808912368, −3.99645783547081037080776123945, −3.46781498153673401324084158170, −1.25855475839182839031994632710, 0.925519101528312968271274583514, 2.03497048869501770690952500278, 3.83707830860855435134033432488, 5.30918706363398688949220857674, 5.54080232761440548633158943046, 6.52174917263171603854151790362, 7.79657303539288709884027990989, 8.425113982733816382512456178135, 9.292018541852774170518375885263, 9.869869428342924919940674984551

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