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Properties

Label 2-117-117.61-c1-0-3
Degree $2$
Conductor $117$
Sign $0.0280 - 0.999i$
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  + (1.32 + 2.30i)2-s + (0.201 − 1.72i)3-s + (−2.52 + 4.37i)4-s + (0.324 + 0.561i)5-s + (4.22 − 1.82i)6-s + 1.54·7-s − 8.11·8-s + (−2.91 − 0.691i)9-s + (−0.861 + 1.49i)10-s + (−2.53 − 4.39i)11-s + (7.02 + 5.22i)12-s + (3.32 + 1.40i)13-s + (2.05 + 3.56i)14-s + (1.03 − 0.445i)15-s + (−5.72 − 9.91i)16-s + (−0.103 − 0.179i)17-s + ⋯
L(s)  = 1  + (0.939 + 1.62i)2-s + (0.116 − 0.993i)3-s + (−1.26 + 2.18i)4-s + (0.145 + 0.251i)5-s + (1.72 − 0.743i)6-s + 0.585·7-s − 2.86·8-s + (−0.973 − 0.230i)9-s + (−0.272 + 0.471i)10-s + (−0.764 − 1.32i)11-s + (2.02 + 1.50i)12-s + (0.921 + 0.389i)13-s + (0.549 + 0.951i)14-s + (0.266 − 0.114i)15-s + (−1.43 − 2.47i)16-s + (−0.0251 − 0.0436i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0280 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0280 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.0280 - 0.999i$
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: \((2,\ 117,\ (\ :1/2),\ 0.0280 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11842 + 1.08751i\)
\(L(\frac12)\) \(\approx\) \(1.11842 + 1.08751i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.201 + 1.72i)T \)
13 \( 1 + (-3.32 - 1.40i)T \)
good2 \( 1 + (-1.32 - 2.30i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.324 - 0.561i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.54T + 7T^{2} \)
11 \( 1 + (2.53 + 4.39i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.103 + 0.179i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.79 + 3.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.20T + 23T^{2} \)
29 \( 1 + (-3.41 - 5.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.58 - 2.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.71 - 8.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.60T + 41T^{2} \)
43 \( 1 - 5.99T + 43T^{2} \)
47 \( 1 + (1.42 - 2.47i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.48T + 53T^{2} \)
59 \( 1 + (-1.49 + 2.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 8.04T + 61T^{2} \)
67 \( 1 - 4.94T + 67T^{2} \)
71 \( 1 + (0.787 + 1.36i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.03T + 73T^{2} \)
79 \( 1 + (-3.23 + 5.60i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.24 + 2.15i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.76 - 3.05i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.38T + 97T^{2} \)
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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

−13.85307903606149962296757696358, −13.31872046759669026037036142786, −12.21650309266245923122898443106, −11.02678698773195248207453640202, −8.495181684505588240432383406914, −8.318148987874524287661727553113, −6.87907688002073586149240780167, −6.16933655185792366269961948530, −5.01364136792540437165862987184, −3.19395390622347273573586790810, 2.15410309416922616765033554958, 3.75646086930831426787544330085, 4.75096760735874456542952818664, 5.68459615276756710760194025168, 8.341102866017034186817793534464, 9.665474464738851343936271044029, 10.35665255240279072064807347076, 11.16294143883441086830335533061, 12.19424471245952644948036465365, 13.13594072719141450475431077035

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