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Properties

Label 2-116-116.11-c1-0-10
Degree $2$
Conductor $116$
Sign $0.916 + 0.400i$
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  + (1.33 − 0.467i)2-s + (1.56 − 1.24i)4-s + (−0.459 + 0.954i)5-s + (1.50 − 2.39i)8-s + (−2.92 + 0.667i)9-s + (−0.167 + 1.48i)10-s + (−0.598 − 0.136i)13-s + (0.890 − 3.89i)16-s + (−3.20 + 3.20i)17-s + (−3.59 + 2.25i)18-s + (0.471 + 2.06i)20-s + (2.41 + 3.03i)25-s + (−0.862 + 0.0971i)26-s + (−4.42 − 3.06i)29-s + (−0.633 − 5.62i)32-s + ⋯
L(s)  = 1  + (0.943 − 0.330i)2-s + (0.781 − 0.623i)4-s + (−0.205 + 0.427i)5-s + (0.532 − 0.846i)8-s + (−0.974 + 0.222i)9-s + (−0.0530 + 0.470i)10-s + (−0.165 − 0.0378i)13-s + (0.222 − 0.974i)16-s + (−0.776 + 0.776i)17-s + (−0.846 + 0.532i)18-s + (0.105 + 0.462i)20-s + (0.483 + 0.606i)25-s + (−0.169 + 0.0190i)26-s + (−0.822 − 0.568i)29-s + (−0.111 − 0.993i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 116,\ (\ :1/2),\ 0.916 + 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56851 - 0.327793i\)
\(L(\frac12)\) \(\approx\) \(1.56851 - 0.327793i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 0.467i)T \)
29 \( 1 + (4.42 + 3.06i)T \)
good3 \( 1 + (2.92 - 0.667i)T^{2} \)
5 \( 1 + (0.459 - 0.954i)T + (-3.11 - 3.90i)T^{2} \)
7 \( 1 + (1.55 + 6.82i)T^{2} \)
11 \( 1 + (4.77 - 9.91i)T^{2} \)
13 \( 1 + (0.598 + 0.136i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (3.20 - 3.20i)T - 17iT^{2} \)
19 \( 1 + (-18.5 - 4.22i)T^{2} \)
23 \( 1 + (-14.3 + 17.9i)T^{2} \)
31 \( 1 + (-24.2 + 19.3i)T^{2} \)
37 \( 1 + (-6.22 + 9.90i)T + (-16.0 - 33.3i)T^{2} \)
41 \( 1 + (-7.65 - 7.65i)T + 41iT^{2} \)
43 \( 1 + (33.6 + 26.8i)T^{2} \)
47 \( 1 + (-20.3 + 42.3i)T^{2} \)
53 \( 1 + (-2.22 - 1.07i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (0.898 + 7.97i)T + (-59.4 + 13.5i)T^{2} \)
67 \( 1 + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-16.1 - 5.64i)T + (57.0 + 45.5i)T^{2} \)
79 \( 1 + (-34.2 - 71.1i)T^{2} \)
83 \( 1 + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (17.8 - 6.23i)T + (69.5 - 55.4i)T^{2} \)
97 \( 1 + (1.11 - 9.90i)T + (-94.5 - 21.5i)T^{2} \)
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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

−13.44456167319160286203918175406, −12.56784432415007285920642012682, −11.29036040297895976561875239560, −10.89115024713352540407553148566, −9.417283207792765080983839813798, −7.86453523847007593267610348196, −6.53071308875761190868357645444, −5.43244471824925107680496813626, −3.93032856291135355177193832970, −2.48892546457847306455162671161, 2.78839801883118608801792578112, 4.38696494319634128534390033520, 5.56482499998316846790818975030, 6.78506179530332544171005550526, 8.068431675902555640080868766871, 9.148086275697083919204274139463, 10.91328183724392913423709749498, 11.75885819567274014798172237048, 12.66193128619498270321056500921, 13.69546536647618769120836137446

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