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Properties

Label 2-1170-13.9-c1-0-0
Degree $2$
Conductor $1170$
Sign $-0.859 + 0.511i$
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 − 5-s + (1 + 1.73i)7-s + 0.999·8-s + (0.5 − 0.866i)10-s + (−2.5 + 4.33i)11-s + (−1 − 3.46i)13-s − 1.99·14-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)17-s + (1 + 1.73i)19-s + (0.499 + 0.866i)20-s + (−2.5 − 4.33i)22-s + (−0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (0.377 + 0.654i)7-s + 0.353·8-s + (0.158 − 0.273i)10-s + (−0.753 + 1.30i)11-s + (−0.277 − 0.960i)13-s − 0.534·14-s + (−0.125 + 0.216i)16-s + (−0.242 − 0.420i)17-s + (0.229 + 0.397i)19-s + (0.111 + 0.193i)20-s + (−0.533 − 0.923i)22-s + (−0.104 + 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.859 + 0.511i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2457263441\)
\(L(\frac12)\) \(\approx\) \(0.2457263441\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + (1 + 3.46i)T \)
good7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 11T + 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.5 - 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8 - 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.00951160785558540183002795556, −9.473859665148937651333239825775, −8.415407784832923272019597645413, −7.75253862867312463914239984826, −7.24109723518160922069449465702, −6.06101002514470061891662554653, −5.17177699702265889004474104333, −4.54621397349729113012302837870, −3.06420844187965613381915352231, −1.82967052604062839174545404150, 0.11725664343122062512449796391, 1.60420868120687565640930258082, 2.96005807308468652889340804570, 3.87873745372381907813598431733, 4.76826011019887716641288391011, 5.86229929031065256797730077483, 7.13387812561495646987171325234, 7.67098409007848145485831192570, 8.723127140811028791652340030552, 9.098965077468531368962901718987

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