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Properties

Label 2-1170-13.3-c1-0-11
Degree $2$
Conductor $1170$
Sign $0.872 - 0.488i$
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 + (1.5 − 2.59i)7-s − 0.999·8-s + (−0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (2.5 + 2.59i)13-s + 3·14-s + (−0.5 − 0.866i)16-s + (2.5 − 4.33i)19-s + (0.499 − 0.866i)20-s + (−0.499 + 0.866i)22-s + (−2 − 3.46i)23-s + 25-s + (−1 + 3.46i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (0.566 − 0.981i)7-s − 0.353·8-s + (−0.158 − 0.273i)10-s + (0.150 + 0.261i)11-s + (0.693 + 0.720i)13-s + 0.801·14-s + (−0.125 − 0.216i)16-s + (0.573 − 0.993i)19-s + (0.111 − 0.193i)20-s + (−0.106 + 0.184i)22-s + (−0.417 − 0.722i)23-s + 0.200·25-s + (−0.196 + 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.872 - 0.488i$
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: \((2,\ 1170,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.956214139\)
\(L(\frac12)\) \(\approx\) \(1.956214139\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 2.59i)T \)
good7 \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9T + 47T^{2} \)
53 \( 1 - 13T + 53T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1 - 1.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 16T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6 - 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.813603328284912531314387128807, −8.755878154409921342509255206246, −8.142434238445517991802423446786, −7.19359523562578291656004608878, −6.73973815426860226526458118268, −5.62251902611953558317987102968, −4.33750342909893239082671336473, −4.23364829268953400896551297506, −2.75657203643884055691848562684, −1.00590953186285163189439512364, 1.13591264955711903338562307158, 2.45877703860876612273764660650, 3.46893217597987582799129750761, 4.37217401115462869967391960978, 5.57547399549434016592645280287, 5.91370061291463132390280694565, 7.35670992402231482243102011618, 8.297591807282520783830513968727, 8.789673843906607229928730563943, 9.878983539633984968969911435725

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