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Properties

Label 2-117-117.110-c1-0-10
Degree $2$
Conductor $117$
Sign $-0.966 + 0.256i$
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  + (0.562 − 2.09i)2-s + (−1.57 − 0.717i)3-s + (−2.35 − 1.35i)4-s + (−0.142 + 0.529i)5-s + (−2.39 + 2.90i)6-s + (−3.03 − 3.03i)7-s + (−1.10 + 1.10i)8-s + (1.97 + 2.26i)9-s + (1.03 + 0.595i)10-s + (3.61 + 0.969i)11-s + (2.73 + 3.83i)12-s + (2.50 − 2.59i)13-s + (−8.06 + 4.65i)14-s + (0.604 − 0.733i)15-s + (−1.02 − 1.77i)16-s + (0.784 + 1.35i)17-s + ⋯
L(s)  = 1  + (0.397 − 1.48i)2-s + (−0.910 − 0.414i)3-s + (−1.17 − 0.679i)4-s + (−0.0635 + 0.237i)5-s + (−0.976 + 1.18i)6-s + (−1.14 − 1.14i)7-s + (−0.389 + 0.389i)8-s + (0.656 + 0.754i)9-s + (0.326 + 0.188i)10-s + (1.09 + 0.292i)11-s + (0.789 + 1.10i)12-s + (0.694 − 0.719i)13-s + (−2.15 + 1.24i)14-s + (0.155 − 0.189i)15-s + (−0.256 − 0.443i)16-s + (0.190 + 0.329i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.966 + 0.256i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.966 + 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.114977 - 0.883211i\)
\(L(\frac12)\) \(\approx\) \(0.114977 - 0.883211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.57 + 0.717i)T \)
13 \( 1 + (-2.50 + 2.59i)T \)
good2 \( 1 + (-0.562 + 2.09i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (0.142 - 0.529i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (3.03 + 3.03i)T + 7iT^{2} \)
11 \( 1 + (-3.61 - 0.969i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.784 - 1.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.27 + 0.876i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 5.03T + 23T^{2} \)
29 \( 1 + (1.02 - 0.593i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.47 + 0.395i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.31 + 0.351i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-3.76 - 3.76i)T + 41iT^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + (1.82 + 6.81i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 5.04iT - 53T^{2} \)
59 \( 1 + (-2.80 - 10.4i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 5.93T + 61T^{2} \)
67 \( 1 + (-2.25 + 2.25i)T - 67iT^{2} \)
71 \( 1 + (0.774 - 2.89i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (9.10 + 9.10i)T + 73iT^{2} \)
79 \( 1 + (-7.18 + 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-13.7 + 3.67i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.63 + 6.09i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.31 + 2.31i)T - 97iT^{2} \)
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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

−13.12825991590670402008838635017, −12.02954403287060479236763634170, −10.90743180601369050631020579634, −10.51217430166102823218567074010, −9.360279443653027490533213843384, −7.24443509617392358682893600792, −6.27753700507350748606955153923, −4.46039529304288451619343806426, −3.31425686627339240584302105395, −1.09518927662725221292613882711, 3.91251047114714827228617894651, 5.28163574451157767498164207305, 6.28955747290940900241178799664, 6.77845538728736641771687700529, 8.736105246396045445599896586751, 9.346379296865148734988878386839, 11.03011209879660746813479866075, 12.19299437751694055051367182074, 13.01152091295700962877733925484, 14.33141301243837676362313387576

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