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Properties

Label 2-1170-195.44-c1-0-9
Degree $2$
Conductor $1170$
Sign $0.383 - 0.923i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
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  + (0.707 − 0.707i)2-s − 1.00i·4-s + (2.23 + 0.137i)5-s + (−3.64 + 3.64i)7-s + (−0.707 − 0.707i)8-s + (1.67 − 1.48i)10-s + (−0.550 + 0.550i)11-s + (−2.57 + 2.52i)13-s + 5.14i·14-s − 1.00·16-s + 3.92i·17-s + (−0.0343 + 0.0343i)19-s + (0.137 − 2.23i)20-s + 0.778i·22-s + 5.25i·23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.998 + 0.0613i)5-s + (−1.37 + 1.37i)7-s + (−0.250 − 0.250i)8-s + (0.529 − 0.468i)10-s + (−0.165 + 0.165i)11-s + (−0.714 + 0.699i)13-s + 1.37i·14-s − 0.250·16-s + 0.951i·17-s + (−0.00788 + 0.00788i)19-s + (0.0306 − 0.499i)20-s + 0.165i·22-s + 1.09i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.383 - 0.923i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.608147611\)
\(L(\frac12)\) \(\approx\) \(1.608147611\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-2.23 - 0.137i)T \)
13 \( 1 + (2.57 - 2.52i)T \)
good7 \( 1 + (3.64 - 3.64i)T - 7iT^{2} \)
11 \( 1 + (0.550 - 0.550i)T - 11iT^{2} \)
17 \( 1 - 3.92iT - 17T^{2} \)
19 \( 1 + (0.0343 - 0.0343i)T - 19iT^{2} \)
23 \( 1 - 5.25iT - 23T^{2} \)
29 \( 1 + 2.61iT - 29T^{2} \)
31 \( 1 + (1.75 - 1.75i)T - 31iT^{2} \)
37 \( 1 + (3.40 - 3.40i)T - 37iT^{2} \)
41 \( 1 + (2.00 + 2.00i)T + 41iT^{2} \)
43 \( 1 - 8.65T + 43T^{2} \)
47 \( 1 + (-3.70 - 3.70i)T + 47iT^{2} \)
53 \( 1 + 7.17T + 53T^{2} \)
59 \( 1 + (6.80 - 6.80i)T - 59iT^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + (9.45 + 9.45i)T + 67iT^{2} \)
71 \( 1 + (-7.43 - 7.43i)T + 71iT^{2} \)
73 \( 1 + (-6.06 + 6.06i)T - 73iT^{2} \)
79 \( 1 + 1.13T + 79T^{2} \)
83 \( 1 + (6.10 - 6.10i)T - 83iT^{2} \)
89 \( 1 + (-5.69 + 5.69i)T - 89iT^{2} \)
97 \( 1 + (0.250 + 0.250i)T + 97iT^{2} \)
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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

−9.788389887934483727970754379926, −9.402848539994797538207018508155, −8.681041649593476960709250925923, −7.19026172090333575002447450735, −6.22350068160950469372132624548, −5.81331662057992539352028241325, −4.90218514606533159777366050747, −3.55103258444018738606595354919, −2.61830140400728014244014810105, −1.84456275480062765596226644891, 0.54935198989425057593209987218, 2.55976717442209992163567182457, 3.40278071184095211324231099516, 4.53236311582198875761039388760, 5.44364617931489084637241654946, 6.34675132365039509814615577697, 6.99172185176013460811678945681, 7.65204178019845911338389454332, 8.924166820382800725353257107300, 9.698674569428722206760883479118

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