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Properties

Label 2-1170-39.20-c1-0-1
Degree $2$
Conductor $1170$
Sign $-0.545 - 0.838i$
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 + (−2.36 − 0.633i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.500i)10-s + (−0.965 + 0.258i)11-s + (3 + 2i)13-s − 2.44i·14-s + (0.500 − 0.866i)16-s + (1.22 + 2.12i)17-s + (−0.464 + 1.73i)19-s + (−0.258 + 0.965i)20-s + (−0.499 − 0.866i)22-s + (−4.50 + 7.79i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.316 − 0.316i)5-s + (−0.894 − 0.239i)7-s + (−0.249 − 0.249i)8-s + (0.273 + 0.158i)10-s + (−0.291 + 0.0780i)11-s + (0.832 + 0.554i)13-s − 0.654i·14-s + (0.125 − 0.216i)16-s + (0.297 + 0.514i)17-s + (−0.106 + 0.397i)19-s + (−0.0578 + 0.215i)20-s + (−0.106 − 0.184i)22-s + (−0.938 + 1.62i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.545 - 0.838i$
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: \((2,\ 1170,\ (\ :1/2),\ -0.545 - 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.306503216\)
\(L(\frac12)\) \(\approx\) \(1.306503216\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 - 2i)T \)
good7 \( 1 + (2.36 + 0.633i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.965 - 0.258i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.22 - 2.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.464 - 1.73i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.50 - 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.24 - 3.60i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.36 - 2.36i)T + 31iT^{2} \)
37 \( 1 + (0.598 + 2.23i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.07 - 7.72i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.13 - 2.96i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.328 - 0.328i)T + 47iT^{2} \)
53 \( 1 - 13.0iT - 53T^{2} \)
59 \( 1 + (0.725 - 2.70i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.83 + 6.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.09 + 1.63i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.74 + 0.466i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-9.92 + 9.92i)T - 73iT^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 + (2.07 - 2.07i)T - 83iT^{2} \)
89 \( 1 + (4.57 - 1.22i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.92 + 7.19i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.870708758173202700514120760693, −9.234429901020191134255960199600, −8.325043456986982422527935232447, −7.60358002389799406907693587593, −6.47682573795502387133413571881, −6.08835709549223384320686127278, −5.05619851215017188544221049109, −3.99666005947741436437315982300, −3.15101094866746958528054504252, −1.44643922434352095059420461852, 0.55326418497386750899340253962, 2.30956441468385574772197214959, 3.05708395158797624124455376984, 4.05896580566821288987362904638, 5.20451016211333867180469914948, 6.14226101089320588641038294313, 6.73951440922243687652848999642, 8.111988377405017897135817762941, 8.748835453389113571328737535431, 9.845797511749170578521386311975

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