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Properties

Label 2-1170-39.20-c1-0-5
Degree $2$
Conductor $1170$
Sign $0.453 - 0.891i$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 + 0.707i)5-s + (0.5 + 0.133i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.500i)10-s + (5.01 − 1.34i)11-s + (0.232 − 3.59i)13-s + 0.517i·14-s + (0.500 − 0.866i)16-s + (1.41 + 2.44i)17-s + (−0.232 + 0.866i)19-s + (0.258 − 0.965i)20-s + (2.59 + 4.50i)22-s + (1.22 − 2.12i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.316 + 0.316i)5-s + (0.188 + 0.0506i)7-s + (−0.249 − 0.249i)8-s + (−0.273 − 0.158i)10-s + (1.51 − 0.405i)11-s + (0.0643 − 0.997i)13-s + 0.138i·14-s + (0.125 − 0.216i)16-s + (0.342 + 0.594i)17-s + (−0.0532 + 0.198i)19-s + (0.0578 − 0.215i)20-s + (0.553 + 0.959i)22-s + (0.255 − 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.453 - 0.891i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.453 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.810643888\)
\(L(\frac12)\) \(\approx\) \(1.810643888\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-0.232 + 3.59i)T \)
good7 \( 1 + (-0.5 - 0.133i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-5.01 + 1.34i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.41 - 2.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.232 - 0.866i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.22 + 2.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.69 - 3.86i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.732 - 0.732i)T + 31iT^{2} \)
37 \( 1 + (0.696 + 2.59i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.31 - 8.62i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.464 - 0.267i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.57 - 4.57i)T + 47iT^{2} \)
53 \( 1 - 1.27iT - 53T^{2} \)
59 \( 1 + (0.896 - 3.34i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.73 - 6.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.46 + 2.53i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.93 + 0.517i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (9.46 - 9.46i)T - 73iT^{2} \)
79 \( 1 + 0.928T + 79T^{2} \)
83 \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \)
89 \( 1 + (-12.4 + 3.32i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2 + 7.46i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.871989785208760208813894426513, −8.815710165226904933860895395196, −8.296603864919878990821396009226, −7.40500428970135436859628322587, −6.50233271717771085975354583943, −5.92294329589463834107576642387, −4.79340883741370065767338526723, −3.85380059718970232968133462160, −2.98313567146216605070673548854, −1.11680205409653030344424747251, 1.00969017967641078258071414659, 2.14334563072664455101405088561, 3.54564420666725123204939099747, 4.30109430176087553903071841603, 5.06131178767096057492496249666, 6.33158684831046685184358860943, 7.06849178740471465159555904752, 8.165866407937019529515676946774, 9.124988712022033437383239698261, 9.484926486587950751791805143301

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