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Properties

Label 2-117-117.59-c1-0-4
Degree $2$
Conductor $117$
Sign $-0.386 - 0.922i$
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  + (−0.754 + 0.754i)2-s + (0.861 + 1.50i)3-s + 0.862i·4-s + (0.595 + 2.22i)5-s + (−1.78 − 0.483i)6-s + (−1.34 − 5.00i)7-s + (−2.15 − 2.15i)8-s + (−1.51 + 2.58i)9-s + (−2.12 − 1.22i)10-s + (1.93 + 1.93i)11-s + (−1.29 + 0.743i)12-s + (3.42 + 1.13i)13-s + (4.78 + 2.76i)14-s + (−2.82 + 2.80i)15-s + 1.52·16-s + (0.0716 + 0.124i)17-s + ⋯
L(s)  = 1  + (−0.533 + 0.533i)2-s + (0.497 + 0.867i)3-s + 0.431i·4-s + (0.266 + 0.993i)5-s + (−0.727 − 0.197i)6-s + (−0.506 − 1.89i)7-s + (−0.763 − 0.763i)8-s + (−0.505 + 0.862i)9-s + (−0.671 − 0.387i)10-s + (0.583 + 0.583i)11-s + (−0.374 + 0.214i)12-s + (0.949 + 0.315i)13-s + (1.27 + 0.737i)14-s + (−0.729 + 0.725i)15-s + 0.382·16-s + (0.0173 + 0.0300i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.512875 + 0.771147i\)
\(L(\frac12)\) \(\approx\) \(0.512875 + 0.771147i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.861 - 1.50i)T \)
13 \( 1 + (-3.42 - 1.13i)T \)
good2 \( 1 + (0.754 - 0.754i)T - 2iT^{2} \)
5 \( 1 + (-0.595 - 2.22i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.34 + 5.00i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.93 - 1.93i)T + 11iT^{2} \)
17 \( 1 + (-0.0716 - 0.124i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.552 + 2.06i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.421 - 0.730i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.605iT - 29T^{2} \)
31 \( 1 + (-3.38 + 0.905i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.973 - 3.63i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (7.55 + 2.02i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.187 - 0.108i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.72 + 10.1i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + 8.00iT - 53T^{2} \)
59 \( 1 + (-5.26 - 5.26i)T + 59iT^{2} \)
61 \( 1 + (-0.675 + 1.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.98 + 7.40i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (7.04 + 1.88i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.76 - 3.76i)T - 73iT^{2} \)
79 \( 1 + (-1.62 - 2.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (14.7 + 3.94i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (14.2 - 3.81i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.525 + 0.140i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.91409582687206723862681798471, −13.28288100563700021712618338641, −11.43812964724415583611798484663, −10.34963493588358526722925164732, −9.743807966769283244867501096207, −8.495493695927505328107910845822, −7.22241714939206423589860825790, −6.59735196518691770750212428303, −4.16052885113244246093033501751, −3.29681143037659109953533805763, 1.40549995222116551758196513686, 2.88934398357885621253360214681, 5.59030180734161502606015864778, 6.21617063047995649081568043890, 8.521711729460237142311823728961, 8.774456793535409894435392715638, 9.659699739088766894812532768496, 11.38144360865399729646568312646, 12.20654539758846547435563157050, 12.94874294446718611082489415511

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