↑

Properties

Label 2-116-116.15-c1-0-7
Degree $2$
Conductor $116$
Sign $-0.197 + 0.980i$
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  + (−1.40 + 0.158i)2-s + (1.94 − 0.445i)4-s + (−3.49 − 2.78i)5-s + (−2.66 + 0.934i)8-s + (1.30 − 2.70i)9-s + (5.35 + 3.36i)10-s + (−2.27 − 4.71i)13-s + (3.60 − 1.73i)16-s + (0.771 + 0.771i)17-s + (−1.40 + 4.00i)18-s + (−8.05 − 3.88i)20-s + (3.33 + 14.6i)25-s + (3.93 + 6.26i)26-s + (3.97 + 3.63i)29-s + (−4.78 + 3.00i)32-s + ⋯
L(s)  = 1  + (−0.993 + 0.111i)2-s + (0.974 − 0.222i)4-s + (−1.56 − 1.24i)5-s + (−0.943 + 0.330i)8-s + (0.433 − 0.900i)9-s + (1.69 + 1.06i)10-s + (−0.629 − 1.30i)13-s + (0.900 − 0.433i)16-s + (0.187 + 0.187i)17-s + (−0.330 + 0.943i)18-s + (−1.80 − 0.867i)20-s + (0.667 + 2.92i)25-s + (0.772 + 1.22i)26-s + (0.737 + 0.675i)29-s + (−0.846 + 0.532i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $-0.197 + 0.980i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 116,\ (\ :1/2),\ -0.197 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.294897 - 0.360080i\)
\(L(\frac12)\) \(\approx\) \(0.294897 - 0.360080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.158i)T \)
29 \( 1 + (-3.97 - 3.63i)T \)
good3 \( 1 + (-1.30 + 2.70i)T^{2} \)
5 \( 1 + (3.49 + 2.78i)T + (1.11 + 4.87i)T^{2} \)
7 \( 1 + (6.30 + 3.03i)T^{2} \)
11 \( 1 + (-8.60 - 6.85i)T^{2} \)
13 \( 1 + (2.27 + 4.71i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 + (-0.771 - 0.771i)T + 17iT^{2} \)
19 \( 1 + (8.24 + 17.1i)T^{2} \)
23 \( 1 + (5.11 - 22.4i)T^{2} \)
31 \( 1 + (-30.2 + 6.89i)T^{2} \)
37 \( 1 + (4.42 - 1.54i)T + (28.9 - 23.0i)T^{2} \)
41 \( 1 + (-8.55 + 8.55i)T - 41iT^{2} \)
43 \( 1 + (41.9 + 9.56i)T^{2} \)
47 \( 1 + (36.7 + 29.3i)T^{2} \)
53 \( 1 + (-5.26 + 6.60i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-11.0 + 6.93i)T + (26.4 - 54.9i)T^{2} \)
67 \( 1 + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-15.1 - 1.70i)T + (71.1 + 16.2i)T^{2} \)
79 \( 1 + (61.7 - 49.2i)T^{2} \)
83 \( 1 + (74.7 - 36.0i)T^{2} \)
89 \( 1 + (4.12 - 0.465i)T + (86.7 - 19.8i)T^{2} \)
97 \( 1 + (16.5 + 10.3i)T + (42.0 + 87.3i)T^{2} \)
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

−12.60277302540870820924846936305, −12.31945395512134720842089268730, −11.19339545437769690263051156600, −9.914412259853906203328832512000, −8.780508111124738381805436633112, −8.012724781467556838795752350293, −7.05080294021387844142354530997, −5.23302671028687135812179589738, −3.57295245009598860154351296116, −0.70837116088211903319368829546, 2.61632574104961542885306975440, 4.21598100667743522999955465605, 6.67714168467182366371515896104, 7.42548792870544209293235191853, 8.232538320086253054059635451684, 9.764957706476724035006102480074, 10.80122934655470994401576711886, 11.48883240712587098583020808629, 12.30609067870448538754278382936, 14.12522452167122602185481828601

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