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Properties

Label 2-1170-13.10-c1-0-2
Degree $2$
Conductor $1170$
Sign $-0.987 - 0.160i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + (4.02 + 2.32i)7-s + 0.999i·8-s + (−0.5 − 0.866i)10-s + (−3.81 + 2.20i)11-s + (−3.35 + 1.32i)13-s − 4.64·14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + (−6.96 − 4.02i)19-s + (0.866 + 0.499i)20-s + (2.20 − 3.81i)22-s + (0.488 + 0.845i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (1.52 + 0.877i)7-s + 0.353i·8-s + (−0.158 − 0.273i)10-s + (−1.14 + 0.663i)11-s + (−0.930 + 0.366i)13-s − 1.24·14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + (−1.59 − 0.922i)19-s + (0.193 + 0.111i)20-s + (0.469 − 0.812i)22-s + (0.101 + 0.176i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.987 - 0.160i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.987 - 0.160i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6761475629\)
\(L(\frac12)\) \(\approx\) \(0.6761475629\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 - iT \)
13 \( 1 + (3.35 - 1.32i)T \)
good7 \( 1 + (-4.02 - 2.32i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.81 - 2.20i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.96 + 4.02i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.488 - 0.845i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.15 + 3.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.44iT - 31T^{2} \)
37 \( 1 + (-3.28 + 1.89i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.31 + 3.64i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.358 + 0.620i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.75iT - 47T^{2} \)
53 \( 1 + 13.5T + 53T^{2} \)
59 \( 1 + (-1.88 - 1.09i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.73 - 6.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.58 - 0.912i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.88 - 3.97i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.36iT - 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 + 3.51iT - 83T^{2} \)
89 \( 1 + (7.07 - 4.08i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.9 - 6.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.24510925062703927345311559763, −9.144682477204073175823826757930, −8.496166064539692554175031891056, −7.77210129448181895759828820659, −7.07297064076272156892025042956, −6.03512634295603291421797410849, −5.06505351764284966415078875966, −4.43756199378839298581800792293, −2.42429505767596539199650804496, −2.00594911879120091550599121157, 0.33267565347274654296874131967, 1.71869645187532397674519956775, 2.78853039780930073150346358858, 4.30770248029770960463042617449, 4.85744026611749095625145626910, 5.98828179482822208783504789613, 7.33959875904368877858659908580, 7.951165523286156957762586401679, 8.349276216469113769416306817851, 9.451416337899354897317805400078

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