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Properties

Label 2-117-117.16-c1-0-1
Degree $2$
Conductor $117$
Sign $0.183 - 0.982i$
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  + 0.697·2-s + (−1.39 + 1.03i)3-s − 1.51·4-s + (1.44 + 2.50i)5-s + (−0.970 + 0.719i)6-s + (1.58 + 2.74i)7-s − 2.45·8-s + (0.874 − 2.86i)9-s + (1.00 + 1.74i)10-s + 2.31·11-s + (2.10 − 1.56i)12-s + (−3.15 − 1.74i)13-s + (1.10 + 1.91i)14-s + (−4.59 − 1.99i)15-s + 1.31·16-s + (2.69 − 4.66i)17-s + ⋯
L(s)  = 1  + 0.493·2-s + (−0.803 + 0.595i)3-s − 0.756·4-s + (0.646 + 1.11i)5-s + (−0.396 + 0.293i)6-s + (0.599 + 1.03i)7-s − 0.866·8-s + (0.291 − 0.956i)9-s + (0.318 + 0.552i)10-s + 0.697·11-s + (0.608 − 0.450i)12-s + (−0.874 − 0.484i)13-s + (0.295 + 0.512i)14-s + (−1.18 − 0.515i)15-s + 0.329·16-s + (0.653 − 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.183 - 0.982i$
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: \((2,\ 117,\ (\ :1/2),\ 0.183 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.761833 + 0.632551i\)
\(L(\frac12)\) \(\approx\) \(0.761833 + 0.632551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.39 - 1.03i)T \)
13 \( 1 + (3.15 + 1.74i)T \)
good2 \( 1 - 0.697T + 2T^{2} \)
5 \( 1 + (-1.44 - 2.50i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.58 - 2.74i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.31T + 11T^{2} \)
17 \( 1 + (-2.69 + 4.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.58 - 4.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.27 + 5.66i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.02T + 29T^{2} \)
31 \( 1 + (-4.23 - 7.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.42 + 4.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.25 - 2.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.99 + 5.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.521 + 0.902i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.29T + 53T^{2} \)
59 \( 1 + 4.70T + 59T^{2} \)
61 \( 1 + (3.71 + 6.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.18 - 7.24i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.680 - 1.17i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 + (0.0365 - 0.0633i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.08 - 1.88i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.0891 + 0.154i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.0654 + 0.113i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.19412466040001184220372859121, −12.42292090811082617369668295782, −11.94901442445934493715780979178, −10.54897938857139967504976339939, −9.769348286686852795793348180147, −8.652399579617752341143177961334, −6.69978673719845899743968349762, −5.63322339433502623327789631150, −4.74573954420172535843065412326, −3.00866071595926337730911290673, 1.24406550412890253253112861637, 4.37380276409098294513563061684, 5.07388655626070758045154180386, 6.33859682859897031095535936400, 7.80080826299422948798428169079, 9.066100829755404837312750539617, 10.16226927791258232991191218000, 11.56501531280469751945436860310, 12.49135812175612724781225974688, 13.33097424246279120825736836058

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