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Properties

Label 2-117-13.4-c1-0-1
Degree $2$
Conductor $117$
Sign $0.711 - 0.702i$
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  + (1.5 + 0.866i)2-s + (0.5 + 0.866i)4-s + 1.73i·5-s − 1.73i·8-s + (−1.49 + 2.59i)10-s + (−2.5 − 2.59i)13-s + (2.49 − 4.33i)16-s + (−1.5 − 2.59i)17-s + (−3 + 1.73i)19-s + (−1.50 + 0.866i)20-s + (−3 + 5.19i)23-s + 2.00·25-s + (−1.5 − 6.06i)26-s + (1.5 − 2.59i)29-s + 3.46i·31-s + (4.5 − 2.59i)32-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s + (0.250 + 0.433i)4-s + 0.774i·5-s − 0.612i·8-s + (−0.474 + 0.821i)10-s + (−0.693 − 0.720i)13-s + (0.624 − 1.08i)16-s + (−0.363 − 0.630i)17-s + (−0.688 + 0.397i)19-s + (−0.335 + 0.193i)20-s + (−0.625 + 1.08i)23-s + 0.400·25-s + (−0.294 − 1.18i)26-s + (0.278 − 0.482i)29-s + 0.622i·31-s + (0.795 − 0.459i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50165 + 0.616372i\)
\(L(\frac12)\) \(\approx\) \(1.50165 + 0.616372i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (2.5 + 2.59i)T \)
good2 \( 1 + (-1.5 - 0.866i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 1.73iT - 5T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (-7.5 - 4.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (6 - 3.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3 - 1.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 1.73i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (-6 - 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.82537752038117811414386352950, −12.91113829320065756769769732237, −11.87492882753469792665319424840, −10.56540634359934074886604029254, −9.578556319675191966040292762700, −7.81619518739241434056211690148, −6.79240429228524573602732606305, −5.75306601814164473145468760627, −4.49171745133503383136198757094, −3.01061525879386457489213948438, 2.30048864547383042155892541730, 4.12193749243753436827426393249, 4.93028819176766711137701447128, 6.38007784589126407921637885043, 8.121361817311082119247377055501, 9.147738395607172997143644102893, 10.60685065900241612146680633728, 11.65594467670936743887824648803, 12.58913598375338648718720352897, 13.09663201674913409202885206212

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