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Properties

Label 2-117-117.16-c1-0-0
Degree $2$
Conductor $117$
Sign $0.228 - 0.973i$
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  − 0.867·2-s + (0.973 + 1.43i)3-s − 1.24·4-s + (−0.0324 − 0.0561i)5-s + (−0.844 − 1.24i)6-s + (1.96 + 3.39i)7-s + 2.81·8-s + (−1.10 + 2.78i)9-s + (0.0281 + 0.0486i)10-s − 5.29·11-s + (−1.21 − 1.78i)12-s + (3.21 + 1.63i)13-s + (−1.70 − 2.94i)14-s + (0.0488 − 0.101i)15-s + 0.0519·16-s + (2.28 − 3.95i)17-s + ⋯
L(s)  = 1  − 0.613·2-s + (0.562 + 0.827i)3-s − 0.623·4-s + (−0.0144 − 0.0251i)5-s + (−0.344 − 0.507i)6-s + (0.741 + 1.28i)7-s + 0.995·8-s + (−0.367 + 0.929i)9-s + (0.00888 + 0.0153i)10-s − 1.59·11-s + (−0.350 − 0.515i)12-s + (0.891 + 0.453i)13-s + (−0.454 − 0.787i)14-s + (0.0126 − 0.0260i)15-s + 0.0129·16-s + (0.553 − 0.958i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.228 - 0.973i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.228 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.639605 + 0.506734i\)
\(L(\frac12)\) \(\approx\) \(0.639605 + 0.506734i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.973 - 1.43i)T \)
13 \( 1 + (-3.21 - 1.63i)T \)
good2 \( 1 + 0.867T + 2T^{2} \)
5 \( 1 + (0.0324 + 0.0561i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.96 - 3.39i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 5.29T + 11T^{2} \)
17 \( 1 + (-2.28 + 3.95i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.281 + 0.486i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.42 + 2.47i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.00T + 29T^{2} \)
31 \( 1 + (4.23 + 7.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.506 + 0.877i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.674 - 1.16i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.45 - 5.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.22 - 3.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.68T + 53T^{2} \)
59 \( 1 - 9.14T + 59T^{2} \)
61 \( 1 + (3.17 + 5.50i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.76 - 3.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.02 - 8.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.39T + 73T^{2} \)
79 \( 1 + (-5.67 + 9.83i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.87 - 3.24i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.00 + 3.47i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.67 + 2.90i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.91256000734946711200028045563, −12.87609473684770034300966968038, −11.37054445177707448415817186086, −10.39482926248915726815917710673, −9.358807396981726584757274749798, −8.506625835633438014102976730112, −7.88142597689981547103809707130, −5.46600655907720470425456724459, −4.58727639312782655362043436279, −2.62029524357180339905401669449, 1.23550503508962710164639809298, 3.59625842300135787915127464184, 5.28117400821768536197199859593, 7.22455522635631882904976715948, 7.983384885353210194882310327616, 8.676091133821957388173899713316, 10.26595387520683290509710313773, 10.84798293078442863586935274542, 12.64408947229659202056872454308, 13.42854703978330744282673077490

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