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Properties

Label 2-117-117.103-c1-0-1
Degree $2$
Conductor $117$
Sign $0.963 - 0.267i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
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  + (−2.14 + 1.23i)2-s + (−1.72 − 0.0890i)3-s + (2.06 − 3.58i)4-s + (−0.771 − 0.445i)5-s + (3.82 − 1.95i)6-s + (0.850 − 0.491i)7-s + 5.29i·8-s + (2.98 + 0.308i)9-s + 2.20·10-s + (2.49 − 1.44i)11-s + (−3.89 + 6.01i)12-s + (0.714 + 3.53i)13-s + (−1.21 + 2.10i)14-s + (1.29 + 0.839i)15-s + (−2.42 − 4.19i)16-s + 5.34·17-s + ⋯
L(s)  = 1  + (−1.51 + 0.875i)2-s + (−0.998 − 0.0514i)3-s + (1.03 − 1.79i)4-s + (−0.344 − 0.199i)5-s + (1.56 − 0.796i)6-s + (0.321 − 0.185i)7-s + 1.87i·8-s + (0.994 + 0.102i)9-s + 0.697·10-s + (0.753 − 0.435i)11-s + (−1.12 + 1.73i)12-s + (0.198 + 0.980i)13-s + (−0.325 + 0.563i)14-s + (0.334 + 0.216i)15-s + (−0.605 − 1.04i)16-s + 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.963 - 0.267i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.963 - 0.267i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.395876 + 0.0539918i\)
\(L(\frac12)\) \(\approx\) \(0.395876 + 0.0539918i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.0890i)T \)
13 \( 1 + (-0.714 - 3.53i)T \)
good2 \( 1 + (2.14 - 1.23i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (0.771 + 0.445i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.850 + 0.491i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.49 + 1.44i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 5.34T + 17T^{2} \)
19 \( 1 + 7.19iT - 19T^{2} \)
23 \( 1 + (-2.31 + 4.01i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.971 + 1.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.73 - 5.04i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.82iT - 37T^{2} \)
41 \( 1 + (-2.39 - 1.38i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.45 + 4.25i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.82 - 2.20i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.30T + 53T^{2} \)
59 \( 1 + (2.74 + 1.58i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.76 - 4.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.15 + 1.81i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.69iT - 71T^{2} \)
73 \( 1 + 9.33iT - 73T^{2} \)
79 \( 1 + (2.37 + 4.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.78 - 2.76i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 17.5iT - 89T^{2} \)
97 \( 1 + (-0.213 + 0.123i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.80863941318914013198472222640, −12.07030980308500995358902459488, −11.27827335515356593233262983878, −10.28055201143600542250362787927, −9.216588975535857393140363961435, −8.166351776585555104655856984365, −6.97018872235351656112585473591, −6.24391227588474004092618787161, −4.68472940201499126399116377241, −1.00995761031371622742523934341, 1.34912532480348257495157277875, 3.61058724618862481099785365288, 5.68884388334778820588357594202, 7.33605391614814609989614427256, 8.147069083524789397428693120095, 9.696855902840885913970807548117, 10.23706379352188641981318111981, 11.39720748911258598988557766925, 11.90275109163402953526819315235, 12.80850948362376915936499850973

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