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Properties

Label 2-117-117.83-c1-0-4
Degree $2$
Conductor $117$
Sign $0.922 - 0.385i$
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  + (−2.24 + 0.600i)2-s + (0.724 + 1.57i)3-s + (2.92 − 1.69i)4-s + (1.01 − 3.78i)5-s + (−2.56 − 3.09i)6-s + (2.27 − 0.608i)7-s + (−2.26 + 2.26i)8-s + (−1.95 + 2.27i)9-s + 9.08i·10-s + (0.637 + 2.38i)11-s + (4.78 + 3.38i)12-s + (1.99 − 3.00i)13-s + (−4.72 + 2.72i)14-s + (6.68 − 1.14i)15-s + (0.335 − 0.581i)16-s + 0.901·17-s + ⋯
L(s)  = 1  + (−1.58 + 0.424i)2-s + (0.418 + 0.908i)3-s + (1.46 − 0.845i)4-s + (0.453 − 1.69i)5-s + (−1.04 − 1.26i)6-s + (0.858 − 0.230i)7-s + (−0.801 + 0.801i)8-s + (−0.650 + 0.759i)9-s + 2.87i·10-s + (0.192 + 0.717i)11-s + (1.38 + 0.976i)12-s + (0.553 − 0.832i)13-s + (−1.26 + 0.729i)14-s + (1.72 − 0.295i)15-s + (0.0838 − 0.145i)16-s + 0.218·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.922 - 0.385i$
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: \((2,\ 117,\ (\ :1/2),\ 0.922 - 0.385i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656818 + 0.131723i\)
\(L(\frac12)\) \(\approx\) \(0.656818 + 0.131723i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.724 - 1.57i)T \)
13 \( 1 + (-1.99 + 3.00i)T \)
good2 \( 1 + (2.24 - 0.600i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-1.01 + 3.78i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.27 + 0.608i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.637 - 2.38i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 0.901T + 17T^{2} \)
19 \( 1 + (2.07 - 2.07i)T - 19iT^{2} \)
23 \( 1 + (-1.50 - 2.60i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.19 - 1.26i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.92 - 0.516i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (7.88 + 7.88i)T + 37iT^{2} \)
41 \( 1 + (0.895 - 3.34i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-5.11 - 2.95i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.259 - 0.966i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 0.635iT - 53T^{2} \)
59 \( 1 + (5.54 + 1.48i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.38 - 7.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.84 + 2.37i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (7.93 + 7.93i)T + 71iT^{2} \)
73 \( 1 + (-9.16 - 9.16i)T + 73iT^{2} \)
79 \( 1 + (0.204 - 0.353i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.66 - 0.982i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (10.0 - 10.0i)T - 89iT^{2} \)
97 \( 1 + (0.733 + 2.73i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.76522553668930932826193898755, −12.46950227425898288849391338489, −10.96529773834811789960097845640, −10.05001158092943499196948095233, −9.169177496443813941043255271024, −8.479533100890136051194788873705, −7.72700055771208149616292663263, −5.65054115754548528664990125883, −4.45964904925987665761220357103, −1.52509719836628135297247262926, 1.81165844178678455103924968030, 2.97951775036803070516497326471, 6.32849185226057388628628728474, 7.11260518813545481899740024845, 8.240945338428186487694289612719, 9.048305871941290891006053587852, 10.39647051415069651888665537103, 11.18966689120698934304052080258, 11.86556846579138698335888656222, 13.73023864220123120683254535369

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