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Properties

Label 2-116-116.107-c0-0-0
Degree $2$
Conductor $116$
Sign $0.938 - 0.345i$
Analytic cond. $0.0578915$
Root an. cond. $0.240606$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (0.400 − 0.193i)5-s + (−0.222 + 0.974i)8-s + (−0.222 + 0.974i)9-s + (−0.277 + 0.347i)10-s + (−0.277 − 1.21i)13-s + (−0.222 − 0.974i)16-s − 1.80·17-s + (−0.222 − 0.974i)18-s + (0.0990 − 0.433i)20-s + (−0.499 + 0.626i)25-s + (0.777 + 0.974i)26-s + (−0.222 − 0.974i)29-s + (0.623 + 0.781i)32-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (0.400 − 0.193i)5-s + (−0.222 + 0.974i)8-s + (−0.222 + 0.974i)9-s + (−0.277 + 0.347i)10-s + (−0.277 − 1.21i)13-s + (−0.222 − 0.974i)16-s − 1.80·17-s + (−0.222 − 0.974i)18-s + (0.0990 − 0.433i)20-s + (−0.499 + 0.626i)25-s + (0.777 + 0.974i)26-s + (−0.222 − 0.974i)29-s + (0.623 + 0.781i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.938 - 0.345i$
Analytic conductor: \(0.0578915\)
Root analytic conductor: \(0.240606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 116,\ (\ :0),\ 0.938 - 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4257392580\)
\(L(\frac12)\) \(\approx\) \(0.4257392580\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 - 0.433i)T \)
29 \( 1 + (0.222 + 0.974i)T \)
good3 \( 1 + (0.222 - 0.974i)T^{2} \)
5 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
7 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.900 - 0.433i)T^{2} \)
13 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + 1.80T + T^{2} \)
19 \( 1 + (0.222 + 0.974i)T^{2} \)
23 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (-0.623 + 0.781i)T^{2} \)
37 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
41 \( 1 - 1.24T + T^{2} \)
43 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.900 - 0.433i)T^{2} \)
53 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
67 \( 1 + (0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
79 \( 1 + (0.900 + 0.433i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
97 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
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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

−13.87815191103923083814203492603, −13.02039422204479340809031940920, −11.40820875245916494763818012549, −10.59506596567294664753546040716, −9.568815941000766169741361940621, −8.449234357076432716096816256328, −7.52818446314234848157134324994, −6.13947680312980219114799289374, −4.99363779169325571996143279210, −2.31827131435495899951581404568, 2.26607837765745910984800191378, 4.05244636358745114329875164559, 6.29445753897241092991800488921, 7.16357847299187705940182612447, 8.837159895492262790052303301048, 9.319946352849133700748968440668, 10.59330318001536347634774278641, 11.54254152352514357558457178863, 12.43466690849536647688375751949, 13.63790628989257083554662323178

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