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Properties

Label 2-1170-13.3-c1-0-9
Degree $2$
Conductor $1170$
Sign $0.189 - 0.981i$
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 + (−0.133 + 0.232i)7-s − 0.999·8-s + (0.5 + 0.866i)10-s + (0.866 + 1.5i)11-s + (3.59 − 0.232i)13-s − 0.267·14-s + (−0.5 − 0.866i)16-s + (3.46 − 6i)17-s + (−2.5 + 4.33i)19-s + (−0.499 + 0.866i)20-s + (−0.866 + 1.5i)22-s + (3.46 + 6i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (−0.0506 + 0.0877i)7-s − 0.353·8-s + (0.158 + 0.273i)10-s + (0.261 + 0.452i)11-s + (0.997 − 0.0643i)13-s − 0.0716·14-s + (−0.125 − 0.216i)16-s + (0.840 − 1.45i)17-s + (−0.573 + 0.993i)19-s + (−0.111 + 0.193i)20-s + (−0.184 + 0.319i)22-s + (0.722 + 1.25i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.189 - 0.981i$
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: \((2,\ 1170,\ (\ :1/2),\ 0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.161939372\)
\(L(\frac12)\) \(\approx\) \(2.161939372\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.59 + 0.232i)T \)
good7 \( 1 + (0.133 - 0.232i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.866 - 1.5i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.46 + 6i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.46 - 6i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.26 - 2.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.535T + 31T^{2} \)
37 \( 1 + (-1.59 - 2.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.46 + 6i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.46 - 7.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.464T + 47T^{2} \)
53 \( 1 - 6.46T + 53T^{2} \)
59 \( 1 + (-1.73 + 3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.26 - 3.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.19 - 7.26i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.46 + 11.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 + 2.53T + 83T^{2} \)
89 \( 1 + (2.59 + 4.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.26 - 9.12i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.754422141302193000832357530535, −9.149394331149546959071247122101, −8.226068319150321311802771951066, −7.37689536100150407235108834902, −6.59515298351058477042895021343, −5.70780572255674013203397299369, −5.04256355289871279398798468818, −3.87759930835034927628670290071, −2.94434984208847641164826232080, −1.39151060773204918536899851507, 0.975403391623490609882584616542, 2.21770361543136625114415028150, 3.38763232211114532699279904191, 4.20612019842646461392983292969, 5.31453435421313718729783401203, 6.17674589446977439656525439806, 6.81636310349755965590393674436, 8.341288629054567333302279244724, 8.722140084559795589364295054293, 9.767907394919590408236384373889

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