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Properties

Label 2-117-117.83-c1-0-7
Degree $2$
Conductor $117$
Sign $0.989 - 0.141i$
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  + (1.85 − 0.496i)2-s + (−0.584 + 1.63i)3-s + (1.45 − 0.837i)4-s + (0.231 − 0.863i)5-s + (−0.274 + 3.30i)6-s + (1.59 − 0.426i)7-s + (−0.439 + 0.439i)8-s + (−2.31 − 1.90i)9-s − 1.71i·10-s + (0.0407 + 0.152i)11-s + (0.517 + 2.85i)12-s + (−2.35 − 2.72i)13-s + (2.73 − 1.58i)14-s + (1.27 + 0.882i)15-s + (−2.27 + 3.93i)16-s − 5.15·17-s + ⋯
L(s)  = 1  + (1.30 − 0.350i)2-s + (−0.337 + 0.941i)3-s + (0.725 − 0.418i)4-s + (0.103 − 0.386i)5-s + (−0.112 + 1.35i)6-s + (0.602 − 0.161i)7-s + (−0.155 + 0.155i)8-s + (−0.771 − 0.635i)9-s − 0.542i·10-s + (0.0122 + 0.0458i)11-s + (0.149 + 0.824i)12-s + (−0.653 − 0.757i)13-s + (0.731 − 0.422i)14-s + (0.328 + 0.227i)15-s + (−0.567 + 0.983i)16-s − 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.989 - 0.141i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.989 - 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69905 + 0.121185i\)
\(L(\frac12)\) \(\approx\) \(1.69905 + 0.121185i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.584 - 1.63i)T \)
13 \( 1 + (2.35 + 2.72i)T \)
good2 \( 1 + (-1.85 + 0.496i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-0.231 + 0.863i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.59 + 0.426i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.0407 - 0.152i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + 5.15T + 17T^{2} \)
19 \( 1 + (-1.74 + 1.74i)T - 19iT^{2} \)
23 \( 1 + (1.40 + 2.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0404 + 0.0233i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.00 - 0.805i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-6.09 - 6.09i)T + 37iT^{2} \)
41 \( 1 + (2.37 - 8.85i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-8.93 - 5.15i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.678 + 2.53i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 9.99iT - 53T^{2} \)
59 \( 1 + (9.43 + 2.52i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.03 + 3.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.82 + 1.56i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-10.2 - 10.2i)T + 71iT^{2} \)
73 \( 1 + (10.2 + 10.2i)T + 73iT^{2} \)
79 \( 1 + (-3.37 + 5.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-12.5 + 3.35i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (7.53 - 7.53i)T - 89iT^{2} \)
97 \( 1 + (2.89 + 10.7i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.49956772863745295701983597071, −12.58799751839095398217828459464, −11.54094550819647441792398566815, −10.83777860552888208890958038107, −9.519558397086968259970886437089, −8.297841605353787773770379051895, −6.32285120338939866778265002943, −4.98487318561799833223657652637, −4.52241660818139798147914546856, −2.91649290188563221584609226529, 2.43272872947529371496719105377, 4.39550140475375367234911394712, 5.60654729888503505602302046149, 6.62397620029546326982318305275, 7.54111490071953538273118192741, 9.099035964059453440012902024408, 10.88701677174189287312098351505, 11.87087273848285873062496098793, 12.56939807819109078513500447523, 13.73155601222285207415944439236

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