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Properties

Label 2-1170-13.3-c1-0-5
Degree $2$
Conductor $1170$
Sign $-0.396 - 0.918i$
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 − 5-s + (0.280 − 0.486i)7-s − 0.999·8-s + (−0.5 − 0.866i)10-s + (2.06 + 3.57i)11-s + (2.84 − 2.21i)13-s + 0.561·14-s + (−0.5 − 0.866i)16-s + (−1.56 + 2.70i)17-s + (−0.280 + 0.486i)19-s + (0.499 − 0.866i)20-s + (−2.06 + 3.57i)22-s + (2.34 + 4.05i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (0.106 − 0.183i)7-s − 0.353·8-s + (−0.158 − 0.273i)10-s + (0.621 + 1.07i)11-s + (0.788 − 0.615i)13-s + 0.150·14-s + (−0.125 − 0.216i)16-s + (−0.378 + 0.655i)17-s + (−0.0644 + 0.111i)19-s + (0.111 − 0.193i)20-s + (−0.439 + 0.761i)22-s + (0.488 + 0.845i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.396 - 0.918i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.396 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.653137457\)
\(L(\frac12)\) \(\approx\) \(1.653137457\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + (-2.84 + 2.21i)T \)
good7 \( 1 + (-0.280 + 0.486i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.06 - 3.57i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.56 - 2.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.280 - 0.486i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.34 - 4.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.21 - 2.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.68T + 31T^{2} \)
37 \( 1 + (-2.06 - 3.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.12 - 10.6i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.219 - 0.379i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7T + 47T^{2} \)
53 \( 1 + 8.56T + 53T^{2} \)
59 \( 1 + (-3.21 + 5.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.12 - 1.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.56 - 11.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 9.36T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 - 7.12T + 83T^{2} \)
89 \( 1 + (9.40 + 16.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.56 + 6.16i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.898281124688745594398235770556, −9.108273501167493489794262187652, −8.210007271306449211244815043923, −7.53634534990167791034305757586, −6.71807241265256384147783675249, −5.91921458071902263260351949777, −4.83485854949387857155647139973, −4.06784591873606981362631175324, −3.15931775262935576494851121271, −1.47827722220413331948394138890, 0.68842120080556210484386696999, 2.15174743755001189161073990169, 3.38033638654084112327032047813, 4.08339876811711341995963889825, 5.10652678720842929781212616922, 6.11531348065321135798161260861, 6.86256810924856419638733953742, 8.045547579788973876134005415196, 8.970154177591302216929990490305, 9.299055363939121806291124595292

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