Properties

Label 2-117-117.11-c1-0-0
Degree $2$
Conductor $117$
Sign $-0.304 + 0.952i$
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  + (−1.86 + 1.86i)2-s + (0.154 + 1.72i)3-s − 4.97i·4-s + (−2.60 − 0.697i)5-s + (−3.50 − 2.93i)6-s + (−2.18 − 0.585i)7-s + (5.54 + 5.54i)8-s + (−2.95 + 0.534i)9-s + (6.16 − 3.55i)10-s + (−1.09 − 1.09i)11-s + (8.57 − 0.769i)12-s + (0.615 + 3.55i)13-s + (5.17 − 2.98i)14-s + (0.800 − 4.60i)15-s − 10.7·16-s + (1.06 − 1.84i)17-s + ⋯
L(s)  = 1  + (−1.32 + 1.32i)2-s + (0.0894 + 0.995i)3-s − 2.48i·4-s + (−1.16 − 0.312i)5-s + (−1.43 − 1.19i)6-s + (−0.826 − 0.221i)7-s + (1.96 + 1.96i)8-s + (−0.984 + 0.178i)9-s + (1.94 − 1.12i)10-s + (−0.330 − 0.330i)11-s + (2.47 − 0.222i)12-s + (0.170 + 0.985i)13-s + (1.38 − 0.798i)14-s + (0.206 − 1.18i)15-s − 2.69·16-s + (0.258 − 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.304 + 0.952i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.304 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0793968 - 0.108677i\)
\(L(\frac12)\) \(\approx\) \(0.0793968 - 0.108677i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.154 - 1.72i)T \)
13 \( 1 + (-0.615 - 3.55i)T \)
good2 \( 1 + (1.86 - 1.86i)T - 2iT^{2} \)
5 \( 1 + (2.60 + 0.697i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (2.18 + 0.585i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.09 + 1.09i)T + 11iT^{2} \)
17 \( 1 + (-1.06 + 1.84i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.84 - 1.02i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.55 - 6.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.234iT - 29T^{2} \)
31 \( 1 + (1.61 - 6.02i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.21 - 0.593i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.19 + 8.18i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.659 - 0.380i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.39 - 0.640i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 3.34iT - 53T^{2} \)
59 \( 1 + (-8.76 - 8.76i)T + 59iT^{2} \)
61 \( 1 + (-4.83 - 8.37i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.39 + 1.98i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.833 - 3.11i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.706 - 0.706i)T - 73iT^{2} \)
79 \( 1 + (-4.39 + 7.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.02 - 3.82i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.560 - 2.09i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.165 - 0.616i)T + (-84.0 - 48.5i)T^{2} \)
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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

−14.79196669049636843399296333338, −13.75117790975456570273680565783, −11.78919724572565314349313107602, −10.67046779394977706776815750184, −9.711385467403797166291944799375, −8.835281258957585348825196182845, −7.992673591792813644215424769911, −6.82320119166668154854354014379, −5.48853705699620088671040456741, −3.96465358882739024261787103015, 0.21114261516731857414099683554, 2.47471435967481940937444027322, 3.64387604055363593676508026709, 6.60847711024006219997877990183, 7.934443733759680335971290285198, 8.292048187181818504890319298516, 9.745057077102303931998446059650, 10.82870796084463752863553151680, 11.66468758937435163034932042571, 12.68591654680258829201352479897

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