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Properties

Label 2-116-116.19-c1-0-4
Degree $2$
Conductor $116$
Sign $0.740 + 0.672i$
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  + (−0.752 − 1.19i)2-s + (−0.867 + 1.80i)4-s + (3.95 + 0.902i)5-s + (2.81 − 0.316i)8-s + (−2.34 − 1.87i)9-s + (−1.89 − 5.41i)10-s + (2.86 − 2.28i)13-s + (−2.49 − 3.12i)16-s + (−4.03 + 4.03i)17-s + (−0.475 + 4.21i)18-s + (−5.05 + 6.34i)20-s + (10.3 + 4.97i)25-s + (−4.89 − 1.71i)26-s + (−5.15 + 1.55i)29-s + (−1.86 + 5.33i)32-s + ⋯
L(s)  = 1  + (−0.532 − 0.846i)2-s + (−0.433 + 0.900i)4-s + (1.76 + 0.403i)5-s + (0.993 − 0.111i)8-s + (−0.781 − 0.623i)9-s + (−0.599 − 1.71i)10-s + (0.795 − 0.634i)13-s + (−0.623 − 0.781i)16-s + (−0.979 + 0.979i)17-s + (−0.111 + 0.993i)18-s + (−1.13 + 1.41i)20-s + (2.06 + 0.994i)25-s + (−0.960 − 0.336i)26-s + (−0.957 + 0.288i)29-s + (−0.330 + 0.943i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.740 + 0.672i$
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: \((2,\ 116,\ (\ :1/2),\ 0.740 + 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.899117 - 0.347500i\)
\(L(\frac12)\) \(\approx\) \(0.899117 - 0.347500i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.752 + 1.19i)T \)
29 \( 1 + (5.15 - 1.55i)T \)
good3 \( 1 + (2.34 + 1.87i)T^{2} \)
5 \( 1 + (-3.95 - 0.902i)T + (4.50 + 2.16i)T^{2} \)
7 \( 1 + (-4.36 + 5.47i)T^{2} \)
11 \( 1 + (-10.7 - 2.44i)T^{2} \)
13 \( 1 + (-2.86 + 2.28i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (4.03 - 4.03i)T - 17iT^{2} \)
19 \( 1 + (-14.8 + 11.8i)T^{2} \)
23 \( 1 + (20.7 - 9.97i)T^{2} \)
31 \( 1 + (13.4 - 27.9i)T^{2} \)
37 \( 1 + (11.9 - 1.34i)T + (36.0 - 8.23i)T^{2} \)
41 \( 1 + (4.80 + 4.80i)T + 41iT^{2} \)
43 \( 1 + (-18.6 - 38.7i)T^{2} \)
47 \( 1 + (45.8 + 10.4i)T^{2} \)
53 \( 1 + (3.23 - 14.1i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-4.90 + 14.0i)T + (-47.6 - 38.0i)T^{2} \)
67 \( 1 + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-1.82 + 2.90i)T + (-31.6 - 65.7i)T^{2} \)
79 \( 1 + (77.0 - 17.5i)T^{2} \)
83 \( 1 + (-51.7 - 64.8i)T^{2} \)
89 \( 1 + (-6.23 - 9.93i)T + (-38.6 + 80.1i)T^{2} \)
97 \( 1 + (-0.530 - 1.51i)T + (-75.8 + 60.4i)T^{2} \)
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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

−13.38211153973663491512384469564, −12.43512812362581268428839617129, −10.99198539144003193934132915931, −10.40023712811952060099619908395, −9.260628535269801081796634580595, −8.548915259689680193004009757000, −6.67549315625065662213632175970, −5.53835775191612896377339106199, −3.38474136554691524730636333324, −1.92041031846691741497683828440, 1.97084430091104261189753726220, 4.97284380328115969489888507867, 5.85628116003536881022428172047, 6.88510378684618745652524867301, 8.584773000998341832020420401946, 9.186392760271150181079740068147, 10.22422845110112928589404868367, 11.29979450181099357688554324313, 13.28166309379293847669716783650, 13.70808805414617982154355714600

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