↑

Properties

Label 2-352-88.69-c1-0-9
Degree $2$
Conductor $352$
Sign $0.829 + 0.559i$
Analytic cond. $2.81073$
Root an. cond. $1.67652$
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  + (2.62 − 0.852i)3-s + (1.70 − 2.35i)5-s + (−0.730 + 2.24i)7-s + (3.72 − 2.70i)9-s + (−2.02 + 2.62i)11-s + (1.49 + 2.05i)13-s + (2.47 − 7.62i)15-s + (−4.77 − 3.46i)17-s + (0.252 − 0.0821i)19-s + 6.51i·21-s − 4.18·23-s + (−1.06 − 3.27i)25-s + (2.60 − 3.58i)27-s + (−3.08 − 1.00i)29-s + (1.66 − 1.20i)31-s + ⋯
L(s)  = 1  + (1.51 − 0.492i)3-s + (0.763 − 1.05i)5-s + (−0.276 + 0.849i)7-s + (1.24 − 0.902i)9-s + (−0.609 + 0.792i)11-s + (0.414 + 0.571i)13-s + (0.639 − 1.96i)15-s + (−1.15 − 0.841i)17-s + (0.0580 − 0.0188i)19-s + 1.42i·21-s − 0.871·23-s + (−0.212 − 0.654i)25-s + (0.500 − 0.689i)27-s + (−0.573 − 0.186i)29-s + (0.298 − 0.217i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.829 + 0.559i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.829 + 0.559i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12049 - 0.648326i\)
\(L(\frac12)\) \(\approx\) \(2.12049 - 0.648326i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (2.02 - 2.62i)T \)
good3 \( 1 + (-2.62 + 0.852i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.70 + 2.35i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.730 - 2.24i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.49 - 2.05i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.77 + 3.46i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.252 + 0.0821i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.18T + 23T^{2} \)
29 \( 1 + (3.08 + 1.00i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.66 + 1.20i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.740 - 0.240i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.927 - 2.85i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.14iT - 43T^{2} \)
47 \( 1 + (2.98 + 9.17i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.77 - 7.94i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-5.50 - 1.78i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.60 + 4.96i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 0.241iT - 67T^{2} \)
71 \( 1 + (3.19 + 2.31i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.96 - 6.04i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.19 - 3.04i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.95 + 12.3i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 16.2T + 89T^{2} \)
97 \( 1 + (-5.90 + 4.29i)T + (29.9 - 92.2i)T^{2} \)
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

−11.66739284801292902470377847160, −9.994814290080342838487945727562, −9.201954348455272815367697659254, −8.823008674356638428614657349858, −7.86212528462603334593609489034, −6.74383944630310251337112664784, −5.47047049541208169219723206118, −4.25394767386274719076506209493, −2.59598944853824244675385087426, −1.85708300933882109647483822960, 2.23288382344851194260233454700, 3.22349253358061459609767067357, 4.07794101290353225831727484526, 5.84918372282719075067619729096, 6.91006121730906886114712630420, 7.993019095499844344006061594710, 8.738078400104865773526066457070, 9.870627845106617380209994634700, 10.42050634650237511566516760911, 11.08322561976670536056240802510

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