Properties

Label 2-116-29.22-c1-0-0
Degree $2$
Conductor $116$
Sign $0.626 - 0.779i$
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  + (−0.376 + 0.781i)3-s + (0.455 + 1.99i)5-s + (0.821 + 0.395i)7-s + (1.40 + 1.75i)9-s + (−1.22 − 0.974i)11-s + (1.33 − 1.67i)13-s + (−1.73 − 0.395i)15-s + 0.772i·17-s + (−3.09 − 6.41i)19-s + (−0.618 + 0.493i)21-s + (0.297 − 1.30i)23-s + (0.722 − 0.347i)25-s + (−4.43 + 1.01i)27-s + (−3.71 − 3.89i)29-s + (0.208 − 0.0476i)31-s + ⋯
L(s)  = 1  + (−0.217 + 0.451i)3-s + (0.203 + 0.893i)5-s + (0.310 + 0.149i)7-s + (0.466 + 0.585i)9-s + (−0.368 − 0.293i)11-s + (0.369 − 0.463i)13-s + (−0.447 − 0.102i)15-s + 0.187i·17-s + (−0.708 − 1.47i)19-s + (−0.134 + 0.107i)21-s + (0.0619 − 0.271i)23-s + (0.144 − 0.0695i)25-s + (−0.854 + 0.194i)27-s + (−0.689 − 0.724i)29-s + (0.0375 − 0.00856i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.626 - 0.779i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 116,\ (\ :1/2),\ 0.626 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.937977 + 0.449466i\)
\(L(\frac12)\) \(\approx\) \(0.937977 + 0.449466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (3.71 + 3.89i)T \)
good3 \( 1 + (0.376 - 0.781i)T + (-1.87 - 2.34i)T^{2} \)
5 \( 1 + (-0.455 - 1.99i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (-0.821 - 0.395i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (1.22 + 0.974i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.33 + 1.67i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 - 0.772iT - 17T^{2} \)
19 \( 1 + (3.09 + 6.41i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (-0.297 + 1.30i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (-0.208 + 0.0476i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (-6.38 + 5.08i)T + (8.23 - 36.0i)T^{2} \)
41 \( 1 - 5.29iT - 41T^{2} \)
43 \( 1 + (4.53 + 1.03i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (-8.07 - 6.43i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (-1.92 - 8.44i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + 1.50T + 59T^{2} \)
61 \( 1 + (-1.46 + 3.04i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (-8.62 - 10.8i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (-4.44 + 5.57i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (7.99 + 1.82i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (-4.00 + 3.19i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (14.2 - 6.86i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (14.2 - 3.25i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (-0.0782 - 0.162i)T + (-60.4 + 75.8i)T^{2} \)
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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

−13.66484228515081397291319769092, −12.83402401152735205395070425425, −11.14339485132796044852985939369, −10.81154871570003454604196155106, −9.678378245395391126501651559762, −8.281574033080145134264294683126, −7.05824450747338565628430005928, −5.75224051028195013818854852137, −4.39325137026806500708453301924, −2.60956188814578885744599924451, 1.53848236613191113262835833547, 4.03859393040755441272666760958, 5.43066358348619521758229923877, 6.72462207809683958571630212708, 7.982563810442587060140353183891, 9.105087918094332883839987023325, 10.19932035564241320922668456336, 11.54424537964015647841318168116, 12.54336265944160344484014592914, 13.12240769518426233682207389000

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