Properties

Label 2-116-116.19-c1-0-0
Degree $2$
Conductor $116$
Sign $0.464 - 0.885i$
Analytic cond. $0.926264$
Root an. cond. $0.962426$
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  + (−1.21 − 0.729i)2-s + (−3.10 − 1.08i)3-s + (0.934 + 1.76i)4-s + (−1.05 − 0.241i)5-s + (2.96 + 3.58i)6-s + (1.22 + 2.54i)7-s + (0.158 − 2.82i)8-s + (6.11 + 4.87i)9-s + (1.10 + 1.06i)10-s + (−0.0781 + 0.693i)11-s + (−0.980 − 6.50i)12-s + (−2.28 + 1.82i)13-s + (0.372 − 3.97i)14-s + (3.01 + 1.89i)15-s + (−2.25 + 3.30i)16-s + (−3.47 + 3.47i)17-s + ⋯
L(s)  = 1  + (−0.856 − 0.516i)2-s + (−1.79 − 0.627i)3-s + (0.467 + 0.884i)4-s + (−0.472 − 0.107i)5-s + (1.21 + 1.46i)6-s + (0.463 + 0.961i)7-s + (0.0560 − 0.998i)8-s + (2.03 + 1.62i)9-s + (0.349 + 0.336i)10-s + (−0.0235 + 0.209i)11-s + (−0.283 − 1.87i)12-s + (−0.634 + 0.506i)13-s + (0.0996 − 1.06i)14-s + (0.779 + 0.489i)15-s + (−0.563 + 0.826i)16-s + (−0.843 + 0.843i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.464 - 0.885i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 116,\ (\ :1/2),\ 0.464 - 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.226114 + 0.136792i\)
\(L(\frac12)\) \(\approx\) \(0.226114 + 0.136792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.21 + 0.729i)T \)
29 \( 1 + (2.03 - 4.98i)T \)
good3 \( 1 + (3.10 + 1.08i)T + (2.34 + 1.87i)T^{2} \)
5 \( 1 + (1.05 + 0.241i)T + (4.50 + 2.16i)T^{2} \)
7 \( 1 + (-1.22 - 2.54i)T + (-4.36 + 5.47i)T^{2} \)
11 \( 1 + (0.0781 - 0.693i)T + (-10.7 - 2.44i)T^{2} \)
13 \( 1 + (2.28 - 1.82i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (3.47 - 3.47i)T - 17iT^{2} \)
19 \( 1 + (-1.15 - 3.31i)T + (-14.8 + 11.8i)T^{2} \)
23 \( 1 + (-2.55 + 0.582i)T + (20.7 - 9.97i)T^{2} \)
31 \( 1 + (1.98 - 1.24i)T + (13.4 - 27.9i)T^{2} \)
37 \( 1 + (-9.64 + 1.08i)T + (36.0 - 8.23i)T^{2} \)
41 \( 1 + (-0.357 - 0.357i)T + 41iT^{2} \)
43 \( 1 + (5.65 - 8.99i)T + (-18.6 - 38.7i)T^{2} \)
47 \( 1 + (4.77 + 0.538i)T + (45.8 + 10.4i)T^{2} \)
53 \( 1 + (0.388 - 1.70i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 - 4.97iT - 59T^{2} \)
61 \( 1 + (-2.11 + 6.04i)T + (-47.6 - 38.0i)T^{2} \)
67 \( 1 + (-2.96 + 3.71i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (4.27 + 5.36i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (3.78 - 6.01i)T + (-31.6 - 65.7i)T^{2} \)
79 \( 1 + (1.34 - 0.151i)T + (77.0 - 17.5i)T^{2} \)
83 \( 1 + (1.23 - 2.55i)T + (-51.7 - 64.8i)T^{2} \)
89 \( 1 + (2.75 + 4.37i)T + (-38.6 + 80.1i)T^{2} \)
97 \( 1 + (-0.776 - 2.21i)T + (-75.8 + 60.4i)T^{2} \)
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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

−12.95769932055520923248200062385, −12.32858623915870025664849313224, −11.56365908921423395458488749231, −10.98757406873536498156676297782, −9.747951566201731020032883392025, −8.251277493476358300001968586884, −7.17139491456634460422655630571, −6.03665589149249538315215639926, −4.58004172343938214477660520996, −1.80090391462244341718682542432, 0.47371697609282356396132075276, 4.45722015432302233914769050002, 5.46359840717920257397438826318, 6.80036848209171667176522968709, 7.59146379098894027434472885817, 9.430614828190463112712805121417, 10.32546110907147686463102974570, 11.28241485560608923859539198271, 11.58880384901774542533727858347, 13.32428615803022581821272482928

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