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Properties

Label 2-117-117.61-c1-0-0
Degree $2$
Conductor $117$
Sign $0.979 + 0.199i$
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  + (−0.900 − 1.56i)2-s + (−1.30 + 1.14i)3-s + (−0.622 + 1.07i)4-s + (1.73 + 2.99i)5-s + (2.95 + 1.00i)6-s + 3.24·7-s − 1.35·8-s + (0.396 − 2.97i)9-s + (3.11 − 5.40i)10-s + (−0.304 − 0.527i)11-s + (−0.419 − 2.11i)12-s + (2.81 + 2.24i)13-s + (−2.92 − 5.06i)14-s + (−5.67 − 1.93i)15-s + (2.46 + 4.27i)16-s + (1.20 + 2.09i)17-s + ⋯
L(s)  = 1  + (−0.636 − 1.10i)2-s + (−0.752 + 0.658i)3-s + (−0.311 + 0.539i)4-s + (0.774 + 1.34i)5-s + (1.20 + 0.410i)6-s + 1.22·7-s − 0.480·8-s + (0.132 − 0.991i)9-s + (0.986 − 1.70i)10-s + (−0.0918 − 0.159i)11-s + (−0.120 − 0.611i)12-s + (0.781 + 0.623i)13-s + (−0.781 − 1.35i)14-s + (−1.46 − 0.499i)15-s + (0.617 + 1.06i)16-s + (0.292 + 0.507i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.979 + 0.199i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.979 + 0.199i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753157 - 0.0760029i\)
\(L(\frac12)\) \(\approx\) \(0.753157 - 0.0760029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.30 - 1.14i)T \)
13 \( 1 + (-2.81 - 2.24i)T \)
good2 \( 1 + (0.900 + 1.56i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.73 - 2.99i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.24T + 7T^{2} \)
11 \( 1 + (0.304 + 0.527i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.20 - 2.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.877 + 1.52i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.162T + 23T^{2} \)
29 \( 1 + (1.45 + 2.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.62 + 8.00i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.826 - 1.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.22T + 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 + (-4.38 + 7.60i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.43T + 53T^{2} \)
59 \( 1 + (-4.13 + 7.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 9.92T + 61T^{2} \)
67 \( 1 - 2.29T + 67T^{2} \)
71 \( 1 + (4.87 + 8.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.59T + 73T^{2} \)
79 \( 1 + (2.53 - 4.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.14 - 7.17i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.93 + 8.55i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.47T + 97T^{2} \)
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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

−13.40732634656896208918829997988, −11.68937331469910795658411808890, −11.27308461920831719095986940886, −10.50132212968976357255698210809, −9.802930966878177244215103039753, −8.553832920263885584491099821087, −6.66915160240827391811658464321, −5.57144686512988356944005777726, −3.69005309721277759654526832232, −1.94470963700072561196067501481, 1.38238773350592815377298134721, 5.10988018191395284226393955348, 5.63402033589812567513996516490, 7.03887208728425594264566026630, 8.199131741862978274437596716827, 8.771701564491761227385575685731, 10.27341473128720374644678021256, 11.65472135856572588647279266829, 12.58332564146429745181873797600, 13.54362669320744526900418994248

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