Properties

Label 2-975-975.584-c0-0-3
Degree $2$
Conductor $975$
Sign $-0.982 + 0.187i$
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  + (1.16 − 1.59i)2-s + (−0.951 + 0.309i)3-s + (−0.896 − 2.76i)4-s + (0.891 − 0.453i)5-s + (−0.610 + 1.87i)6-s + (−3.57 − 1.16i)8-s + (0.809 − 0.587i)9-s + (0.309 − 1.95i)10-s + (−0.734 − 0.533i)11-s + (1.70 + 2.34i)12-s + (0.587 + 0.809i)13-s + (−0.707 + 0.707i)15-s + (−3.65 + 2.65i)16-s − 1.97i·18-s + (−2.05 − 2.05i)20-s + ⋯
L(s)  = 1  + (1.16 − 1.59i)2-s + (−0.951 + 0.309i)3-s + (−0.896 − 2.76i)4-s + (0.891 − 0.453i)5-s + (−0.610 + 1.87i)6-s + (−3.57 − 1.16i)8-s + (0.809 − 0.587i)9-s + (0.309 − 1.95i)10-s + (−0.734 − 0.533i)11-s + (1.70 + 2.34i)12-s + (0.587 + 0.809i)13-s + (−0.707 + 0.707i)15-s + (−3.65 + 2.65i)16-s − 1.97i·18-s + (−2.05 − 2.05i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.982 + 0.187i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (584, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ -0.982 + 0.187i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.368961702\)
\(L(\frac12)\) \(\approx\) \(1.368961702\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 + (-0.891 + 0.453i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
good2 \( 1 + (-1.16 + 1.59i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.734 + 0.533i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.253 + 0.183i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.17iT - T^{2} \)
47 \( 1 + (-0.297 + 0.0966i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-1.44 + 1.04i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.280 + 0.863i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-1.87 - 0.610i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (1.14 + 0.831i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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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

−10.08831755130948594678206772602, −9.558604571349753043148508234650, −8.677839925449933497985150915828, −6.59121833337239780133057247212, −5.92025745519899664545955788793, −5.22850951563441828336108738505, −4.53031245259855072937535513369, −3.55054489984428286056153619092, −2.25822554770458533678005329018, −1.10713423397349020545853462097, 2.52404999577901244857108114913, 3.82242670165022854187933921397, 5.05381044362367564822996725311, 5.48028018714523120290135158470, 6.24496009572841904031230285485, 6.93029784134682791485207693416, 7.63586578555980103540946805384, 8.500824382703307281148769698521, 9.730871595444087969595038185905, 10.68389592298953473301666234419

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