Properties

Label 2-117-13.4-c1-0-2
Degree $2$
Conductor $117$
Sign $0.729 + 0.684i$
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 − 1.73i)4-s + (4.5 − 2.59i)7-s + (−2.5 + 2.59i)13-s + (−1.99 + 3.46i)16-s + (−3 + 1.73i)19-s + 5·25-s + (−9 − 5.19i)28-s + 8.66i·31-s + (−6 − 3.46i)37-s + (6.5 + 11.2i)43-s + (10 − 17.3i)49-s + (7 + 1.73i)52-s + (−6.5 − 11.2i)61-s + 7.99·64-s + (−10.5 − 6.06i)67-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (1.70 − 0.981i)7-s + (−0.693 + 0.720i)13-s + (−0.499 + 0.866i)16-s + (−0.688 + 0.397i)19-s + 25-s + (−1.70 − 0.981i)28-s + 1.55i·31-s + (−0.986 − 0.569i)37-s + (0.991 + 1.71i)43-s + (1.42 − 2.47i)49-s + (0.970 + 0.240i)52-s + (−0.832 − 1.44i)61-s + 0.999·64-s + (−1.28 − 0.740i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.729 + 0.684i$
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: \((2,\ 117,\ (\ :1/2),\ 0.729 + 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.977235 - 0.386551i\)
\(L(\frac12)\) \(\approx\) \(0.977235 - 0.386551i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + (-4.5 + 2.59i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.66iT - 31T^{2} \)
37 \( 1 + (6 + 3.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.5 - 11.2i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.5 + 6.06i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 - 13T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.5 - 9.52i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.81279173160257393401615679087, −12.37128003121127970556062776597, −11.01560413161178701888136302047, −10.50640332215871967047634143278, −9.162552205587415278270033255433, −8.047159372711070323146310603239, −6.78510443580637797647023510823, −5.10354134532696297849656395253, −4.35334440313498098980548492336, −1.56257140765442249388862088029, 2.51377898048347983367857045794, 4.44536589169225904437065558856, 5.43648123126854160264107828925, 7.41249293760598380020186165924, 8.328349868159648652891769166710, 9.067439377155730272574142174468, 10.69394454445487923535021362642, 11.80718845771292300073660906113, 12.46427648873793078139292504378, 13.62694379516947677042840371391

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