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Properties

Label 2-1170-39.20-c1-0-0
Degree $2$
Conductor $1170$
Sign $0.739 - 0.673i$
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 + (−0.866 − 0.232i)7-s + (0.707 + 0.707i)8-s + (0.866 + 0.500i)10-s + (0.965 − 0.258i)11-s + (−2.59 − 2.5i)13-s + 0.896i·14-s + (0.500 − 0.866i)16-s + (2.44 + 4.24i)17-s + (0.232 − 0.866i)19-s + (0.258 − 0.965i)20-s + (−0.499 − 0.866i)22-s + (−2.63 + 4.57i)23-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.316 + 0.316i)5-s + (−0.327 − 0.0877i)7-s + (0.249 + 0.249i)8-s + (0.273 + 0.158i)10-s + (0.291 − 0.0780i)11-s + (−0.720 − 0.693i)13-s + 0.239i·14-s + (0.125 − 0.216i)16-s + (0.594 + 1.02i)17-s + (0.0532 − 0.198i)19-s + (0.0578 − 0.215i)20-s + (−0.106 − 0.184i)22-s + (−0.550 + 0.953i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.739 - 0.673i$
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: \((2,\ 1170,\ (\ :1/2),\ 0.739 - 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9376305955\)
\(L(\frac12)\) \(\approx\) \(0.9376305955\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 + 2.5i)T \)
good7 \( 1 + (0.866 + 0.232i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.965 + 0.258i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.44 - 4.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.232 + 0.866i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.63 - 4.57i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.44 - 1.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.26 - 1.26i)T + 31iT^{2} \)
37 \( 1 + (-1.59 - 5.96i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.03 - 3.86i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (3.92 - 2.26i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.02 - 7.02i)T + 47iT^{2} \)
53 \( 1 - 12.8iT - 53T^{2} \)
59 \( 1 + (1.03 - 3.86i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.46 - 4.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.1 + 2.73i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.93 + 0.517i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.53 + 2.53i)T - 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (9.52 - 9.52i)T - 83iT^{2} \)
89 \( 1 + (-11.7 + 3.13i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.66 - 6.19i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.00958923911646743494077840534, −9.262486410103781550396371866222, −8.185932178586895159652171689834, −7.65233748805171135806683077872, −6.56835710059850263441319014644, −5.60542317932547585667755874496, −4.48740808271982416032736739495, −3.50796069710053713692230379261, −2.72229850296745030972778111380, −1.25331293636386759681620286318, 0.47924010692279843059162003047, 2.24475715871685227478701317347, 3.68364349526916442537587432784, 4.63756896101858131777521553038, 5.44503428472810974906064554535, 6.51169959930105965607468207622, 7.16597151326869203433986170013, 8.009505544477871863721915845397, 8.816931419384557386258770231880, 9.605486288993602868764541875149

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