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Properties

Label 2-1170-65.4-c1-0-13
Degree $2$
Conductor $1170$
Sign $0.683 - 0.730i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.230 + 2.22i)5-s + (−0.432 − 0.749i)7-s + 0.999·8-s + (−1.81 − 1.31i)10-s + (−0.151 − 0.0874i)11-s + (−1.35 − 3.34i)13-s + 0.865·14-s + (−0.5 + 0.866i)16-s + (7.08 − 4.08i)17-s + (5.20 − 3.00i)19-s + (2.04 − 0.912i)20-s + (0.151 − 0.0874i)22-s + (2.52 + 1.45i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.103 + 0.994i)5-s + (−0.163 − 0.283i)7-s + 0.353·8-s + (−0.572 − 0.414i)10-s + (−0.0456 − 0.0263i)11-s + (−0.375 − 0.926i)13-s + 0.231·14-s + (−0.125 + 0.216i)16-s + (1.71 − 0.991i)17-s + (1.19 − 0.689i)19-s + (0.456 − 0.204i)20-s + (0.0322 − 0.0186i)22-s + (0.525 + 0.303i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.683 - 0.730i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.683 - 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.302233700\)
\(L(\frac12)\) \(\approx\) \(1.302233700\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (0.230 - 2.22i)T \)
13 \( 1 + (1.35 + 3.34i)T \)
good7 \( 1 + (0.432 + 0.749i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.151 + 0.0874i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-7.08 + 4.08i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.20 + 3.00i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.52 - 1.45i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.24 - 5.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.95iT - 31T^{2} \)
37 \( 1 + (-0.879 + 1.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.08 - 4.08i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.94 + 4.58i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 2.48iT - 53T^{2} \)
59 \( 1 + (-6.09 + 3.51i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.98 - 6.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.36 - 2.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.2 - 7.08i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 9.48T + 79T^{2} \)
83 \( 1 - 0.139T + 83T^{2} \)
89 \( 1 + (-11.3 - 6.56i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.32 - 7.48i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.845968185952774603085543921790, −9.185340349844614380060998849174, −7.935798235331537344126753088661, −7.35441979444720853597462968218, −6.88952927501411945079144022317, −5.61533091999786919462965133447, −5.11578776948308342825456985851, −3.50641203756179497083227463010, −2.83617440901365159023148120406, −0.919722955755061711518566904380, 0.959797651419678292825180786214, 2.08436633885495748729726412004, 3.46543472213599111382355000596, 4.30409616510728037624425523568, 5.36677905078872230714305225265, 6.15416836511072243630093699444, 7.76251336873891950334476714056, 7.88339275237030653442091486154, 9.199508227134224463178163111818, 9.504512003074996244489688236545

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