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Properties

Label 2-1170-65.4-c1-0-20
Degree $2$
Conductor $1170$
Sign $0.410 + 0.911i$
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 + (−2.10 + 0.750i)5-s + (−0.702 − 1.21i)7-s − 0.999·8-s + (−0.403 + 2.19i)10-s + (4.59 + 2.65i)11-s + (−1.58 + 3.23i)13-s − 1.40·14-s + (−0.5 + 0.866i)16-s + (6.09 − 3.52i)17-s + (2.28 − 1.32i)19-s + (1.70 + 1.44i)20-s + (4.59 − 2.65i)22-s + (−2.30 − 1.33i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.942 + 0.335i)5-s + (−0.265 − 0.460i)7-s − 0.353·8-s + (−0.127 + 0.695i)10-s + (1.38 + 0.800i)11-s + (−0.440 + 0.897i)13-s − 0.375·14-s + (−0.125 + 0.216i)16-s + (1.47 − 0.853i)17-s + (0.524 − 0.303i)19-s + (0.380 + 0.324i)20-s + (0.980 − 0.565i)22-s + (−0.481 − 0.278i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.410 + 0.911i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.410 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.642702845\)
\(L(\frac12)\) \(\approx\) \(1.642702845\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 - 0.750i)T \)
13 \( 1 + (1.58 - 3.23i)T \)
good7 \( 1 + (0.702 + 1.21i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.59 - 2.65i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-6.09 + 3.52i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.28 + 1.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.30 + 1.33i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.10 + 3.64i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.07iT - 31T^{2} \)
37 \( 1 + (-0.412 + 0.715i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.43 + 1.40i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.57 + 2.64i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.79T + 47T^{2} \)
53 \( 1 + 7.93iT - 53T^{2} \)
59 \( 1 + (5.59 - 3.23i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.36 - 12.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.34 + 4.81i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.04T + 73T^{2} \)
79 \( 1 - 4.05T + 79T^{2} \)
83 \( 1 - 8.55T + 83T^{2} \)
89 \( 1 + (-13.1 - 7.57i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.18 + 3.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.721578191747798312766189183735, −9.108998352997535282320715039561, −7.83163796840009159641598856592, −7.12208009010035619459960818793, −6.42081906286110831776462140368, −5.06728065414695043203049890822, −4.12636455782982890808399401202, −3.61920740429055048376610833522, −2.35342864395694242614242475821, −0.841348094394824368203268719794, 1.09096764707361508429884282699, 3.25982906638136678332267315494, 3.64466619222434603006327369693, 4.89297912301718333867418787560, 5.74257200946499717022851001643, 6.49100556386472830773884417065, 7.59638668744690698179694813836, 8.137894971662677102637445719152, 8.906769224881392628981777794500, 9.733467984480954416291183205731

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