↑

Properties

Label 2-116-29.24-c1-0-0
Degree $2$
Conductor $116$
Sign $0.332 - 0.943i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.588 + 2.58i)3-s + (−0.379 − 0.475i)5-s + (0.143 + 0.630i)7-s + (−3.60 + 1.73i)9-s + (2.86 + 1.37i)11-s + (−4.97 − 2.39i)13-s + (1.00 − 1.25i)15-s + 6.66·17-s + (1.14 − 5.02i)19-s + (−1.54 + 0.742i)21-s + (1.01 − 1.27i)23-s + (1.03 − 4.51i)25-s + (−1.65 − 2.08i)27-s + (−5.38 + 0.122i)29-s + (−1.23 − 1.54i)31-s + ⋯
L(s)  = 1  + (0.340 + 1.48i)3-s + (−0.169 − 0.212i)5-s + (0.0544 + 0.238i)7-s + (−1.20 + 0.579i)9-s + (0.863 + 0.415i)11-s + (−1.37 − 0.664i)13-s + (0.259 − 0.325i)15-s + 1.61·17-s + (0.262 − 1.15i)19-s + (−0.336 + 0.162i)21-s + (0.211 − 0.265i)23-s + (0.206 − 0.902i)25-s + (−0.319 − 0.400i)27-s + (−0.999 + 0.0226i)29-s + (−0.221 − 0.277i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.332 - 0.943i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 116,\ (\ :1/2),\ 0.332 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.920018 + 0.651283i\)
\(L(\frac12)\) \(\approx\) \(0.920018 + 0.651283i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.38 - 0.122i)T \)
good3 \( 1 + (-0.588 - 2.58i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (0.379 + 0.475i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (-0.143 - 0.630i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (-2.86 - 1.37i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (4.97 + 2.39i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 - 6.66T + 17T^{2} \)
19 \( 1 + (-1.14 + 5.02i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (-1.01 + 1.27i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (1.23 + 1.54i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (2.60 - 1.25i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + 9.83T + 41T^{2} \)
43 \( 1 + (4.81 - 6.04i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-7.49 - 3.61i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-4.82 - 6.05i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + 1.12T + 59T^{2} \)
61 \( 1 + (2.02 + 8.85i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (11.2 - 5.42i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-0.344 - 0.165i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-0.200 + 0.250i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + (15.5 - 7.46i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 + (1.91 - 8.39i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (5.99 + 7.51i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (-1.95 + 8.56i)T + (-87.3 - 42.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.15906019297343224947635640194, −12.55607819908128685685049287171, −11.64958639859171303168326599123, −10.28434268190448235075191660809, −9.659829616828704386905523312871, −8.701929995565410148744078901157, −7.32602151187432309692594152223, −5.39850615359110643922123513494, −4.43772895864940991528890152682, −3.04659356474381114692215357095, 1.62649923129061217627517206619, 3.46354064703133893143755187244, 5.62512948390626350713036766625, 7.08050560975159264703435246568, 7.54032610327219811266182545257, 8.863210735976551177042467678812, 10.15191452186472313903367142981, 11.87522369886409150422479371865, 12.12165386986212495239156780142, 13.41195986842536371728567642850

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