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Properties

Label 2-116-116.83-c0-0-0
Degree $2$
Conductor $116$
Sign $0.218 - 0.975i$
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  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.277 + 1.21i)5-s + (0.623 − 0.781i)8-s + (0.623 − 0.781i)9-s + (−1.12 − 0.541i)10-s + (−1.12 − 1.40i)13-s + (0.623 + 0.781i)16-s − 0.445·17-s + (0.623 + 0.781i)18-s + (0.777 − 0.974i)20-s + (−0.499 − 0.240i)25-s + (1.62 − 0.781i)26-s + (0.623 + 0.781i)29-s + (−0.900 + 0.433i)32-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.277 + 1.21i)5-s + (0.623 − 0.781i)8-s + (0.623 − 0.781i)9-s + (−1.12 − 0.541i)10-s + (−1.12 − 1.40i)13-s + (0.623 + 0.781i)16-s − 0.445·17-s + (0.623 + 0.781i)18-s + (0.777 − 0.974i)20-s + (−0.499 − 0.240i)25-s + (1.62 − 0.781i)26-s + (0.623 + 0.781i)29-s + (−0.900 + 0.433i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.218 - 0.975i$
Analytic conductor: \(0.0578915\)
Root analytic conductor: \(0.240606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 116,\ (\ :0),\ 0.218 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4923534836\)
\(L(\frac12)\) \(\approx\) \(0.4923534836\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 - 0.974i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
good3 \( 1 + (-0.623 + 0.781i)T^{2} \)
5 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
7 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (0.222 - 0.974i)T^{2} \)
13 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + 0.445T + T^{2} \)
19 \( 1 + (-0.623 - 0.781i)T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.900 + 0.433i)T^{2} \)
37 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
41 \( 1 + 1.80T + T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.222 - 0.974i)T^{2} \)
53 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
67 \( 1 + (0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \)
79 \( 1 + (0.222 + 0.974i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
97 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
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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

−14.40106795523275260690839766768, −13.20258346073359923438809195216, −12.08940785886430973806973782692, −10.48640617364733001908786851926, −9.915717125772257711920818893238, −8.441326966483433793615787218850, −7.22375586503258266377073054359, −6.61775284997599411631168994524, −5.07114890143137279357542216185, −3.36832508695949639784764112068, 1.98953106651989121279289648790, 4.25834907118962172763883245254, 4.98198964292263886909688124692, 7.30024813219237327687015242778, 8.527768779861998348410218713973, 9.381670556886473611805034923313, 10.42584065434807916780992551585, 11.72552978036225993307987305579, 12.37500403247397481488490896898, 13.32241310806183552707423500066

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