↑

Properties

Label 2-116-29.25-c1-0-0
Degree $2$
Conductor $116$
Sign $-0.612 - 0.790i$
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  + (−2.04 + 0.985i)3-s + (−0.299 + 1.31i)5-s + (−3.84 + 1.85i)7-s + (1.34 − 1.68i)9-s + (1.30 + 1.63i)11-s + (0.703 + 0.882i)13-s + (−0.680 − 2.98i)15-s − 1.88·17-s + (4.62 + 2.22i)19-s + (6.04 − 7.57i)21-s + (−1.77 − 7.79i)23-s + (2.87 + 1.38i)25-s + (0.427 − 1.87i)27-s + (5.38 + 0.0974i)29-s + (−2.33 + 10.2i)31-s + ⋯
L(s)  = 1  + (−1.18 + 0.568i)3-s + (−0.133 + 0.587i)5-s + (−1.45 + 0.700i)7-s + (0.447 − 0.561i)9-s + (0.393 + 0.492i)11-s + (0.195 + 0.244i)13-s + (−0.175 − 0.769i)15-s − 0.457·17-s + (1.06 + 0.511i)19-s + (1.31 − 1.65i)21-s + (−0.370 − 1.62i)23-s + (0.574 + 0.276i)25-s + (0.0822 − 0.360i)27-s + (0.999 + 0.0181i)29-s + (−0.419 + 1.83i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $-0.612 - 0.790i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 116,\ (\ :1/2),\ -0.612 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.225226 + 0.459302i\)
\(L(\frac12)\) \(\approx\) \(0.225226 + 0.459302i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.38 - 0.0974i)T \)
good3 \( 1 + (2.04 - 0.985i)T + (1.87 - 2.34i)T^{2} \)
5 \( 1 + (0.299 - 1.31i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (3.84 - 1.85i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (-1.30 - 1.63i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.703 - 0.882i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 1.88T + 17T^{2} \)
19 \( 1 + (-4.62 - 2.22i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (1.77 + 7.79i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (2.33 - 10.2i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (3.28 - 4.12i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + 8.84T + 41T^{2} \)
43 \( 1 + (-1.02 - 4.49i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-4.31 - 5.41i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (0.592 - 2.59i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 - 5.29T + 59T^{2} \)
61 \( 1 + (4.27 - 2.05i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (-1.32 + 1.66i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (10.2 + 12.9i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-1.28 - 5.63i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (-7.06 + 8.86i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (-11.6 - 5.61i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-1.21 + 5.33i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-2.98 - 1.43i)T + (60.4 + 75.8i)T^{2} \)
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

−13.94955210029996829663556535596, −12.45582232742513933654583071825, −11.96743233497636372904471493459, −10.67960122505802034108206962973, −10.00383698060306877231512231130, −8.838906742658072372944703050873, −6.81990862840286675946928091962, −6.19451756854909158306816778147, −4.81853135719941026600209521222, −3.15464975096016236189996000375, 0.64884706385576653599664055070, 3.63384373549972487740596833047, 5.39934147273872094407569685252, 6.42666749669481732148417404329, 7.34485053165994363261388929472, 9.023944129295130697768245228425, 10.12530890738715239520123436445, 11.34710288266290222120306663715, 12.12676333074694968950221402421, 13.14351323827968775386097053669

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