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Properties

Label 2-117-117.4-c1-0-10
Degree $2$
Conductor $117$
Sign $-0.932 + 0.362i$
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  − 1.40i·2-s + (−1.72 + 0.172i)3-s + 0.0239·4-s + (−2.61 − 1.50i)5-s + (0.242 + 2.42i)6-s + (−2.76 − 1.59i)7-s − 2.84i·8-s + (2.94 − 0.594i)9-s + (−2.12 + 3.67i)10-s − 0.793i·11-s + (−0.0411 + 0.00411i)12-s + (1.75 + 3.15i)13-s + (−2.24 + 3.88i)14-s + (4.76 + 2.15i)15-s − 3.95·16-s + (−0.0957 − 0.165i)17-s + ⋯
L(s)  = 1  − 0.994i·2-s + (−0.995 + 0.0994i)3-s + 0.0119·4-s + (−1.16 − 0.675i)5-s + (0.0988 + 0.989i)6-s + (−1.04 − 0.603i)7-s − 1.00i·8-s + (0.980 − 0.198i)9-s + (−0.671 + 1.16i)10-s − 0.239i·11-s + (−0.0118 + 0.00118i)12-s + (0.486 + 0.873i)13-s + (−0.600 + 1.03i)14-s + (1.23 + 0.555i)15-s − 0.987·16-s + (−0.0232 − 0.0402i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.932 + 0.362i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.932 + 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.108920 - 0.580618i\)
\(L(\frac12)\) \(\approx\) \(0.108920 - 0.580618i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.172i)T \)
13 \( 1 + (-1.75 - 3.15i)T \)
good2 \( 1 + 1.40iT - 2T^{2} \)
5 \( 1 + (2.61 + 1.50i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.76 + 1.59i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.793iT - 11T^{2} \)
17 \( 1 + (0.0957 + 0.165i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.28 + 3.62i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.27 + 2.20i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.44T + 29T^{2} \)
31 \( 1 + (6.22 + 3.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.25 - 1.30i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.784 - 0.452i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.98 + 3.44i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.60 - 3.81i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.0692T + 53T^{2} \)
59 \( 1 - 13.5iT - 59T^{2} \)
61 \( 1 + (0.0894 - 0.154i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.17 + 5.29i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.271 + 0.156i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.55iT - 73T^{2} \)
79 \( 1 + (2.13 + 3.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.01 + 4.04i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.56 + 2.63i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-15.2 - 8.78i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.66822608243508941930351133925, −11.84177219862928340780869079548, −11.29661957691641749221102584757, −10.22833497410312427050612693158, −9.206169998686923361548859567868, −7.38597972599709721996922867729, −6.41361943691710901887243212617, −4.52530895381128800350442114869, −3.50706878986142189738643680269, −0.74302861197515973240124575278, 3.36460715688102696208790051365, 5.33146795201786886581353160625, 6.31915733035532732824638654994, 7.21769691776345746087935132591, 8.113250451412053817921222292817, 9.899233591480075285062545165627, 11.07992594248041554606573813919, 11.87933299618893542183363163434, 12.75210113780029472304684354488, 14.29756411086671701994219785967

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