Properties

Label 2-352-88.5-c1-0-5
Degree $2$
Conductor $352$
Sign $0.138 + 0.990i$
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  + (−1.67 + 2.31i)3-s + (−2.48 − 0.805i)5-s + (0.158 − 0.115i)7-s + (−1.59 − 4.90i)9-s + (−2.89 − 1.62i)11-s + (3.78 − 1.22i)13-s + (6.02 − 4.37i)15-s + (−0.382 + 1.17i)17-s + (3.21 − 4.42i)19-s + 0.560i·21-s − 3.19·23-s + (1.45 + 1.05i)25-s + (5.87 + 1.90i)27-s + (1.78 + 2.45i)29-s + (−3.03 − 9.35i)31-s + ⋯
L(s)  = 1  + (−0.969 + 1.33i)3-s + (−1.10 − 0.360i)5-s + (0.0600 − 0.0436i)7-s + (−0.531 − 1.63i)9-s + (−0.871 − 0.490i)11-s + (1.04 − 0.340i)13-s + (1.55 − 1.13i)15-s + (−0.0928 + 0.285i)17-s + (0.738 − 1.01i)19-s + 0.122i·21-s − 0.666·23-s + (0.291 + 0.211i)25-s + (1.13 + 0.367i)27-s + (0.331 + 0.455i)29-s + (−0.545 − 1.67i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.138 + 0.990i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.138 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.253211 - 0.220371i\)
\(L(\frac12)\) \(\approx\) \(0.253211 - 0.220371i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.89 + 1.62i)T \)
good3 \( 1 + (1.67 - 2.31i)T + (-0.927 - 2.85i)T^{2} \)
5 \( 1 + (2.48 + 0.805i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.158 + 0.115i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-3.78 + 1.22i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.382 - 1.17i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.21 + 4.42i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.19T + 23T^{2} \)
29 \( 1 + (-1.78 - 2.45i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.03 + 9.35i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.43 + 6.10i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.91 + 4.30i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.82iT - 43T^{2} \)
47 \( 1 + (0.410 + 0.298i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.0202 + 0.00659i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.31 + 4.56i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (12.3 + 4.00i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 2.16iT - 67T^{2} \)
71 \( 1 + (1.31 - 4.06i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.07 + 1.50i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.57 - 4.85i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.18 - 0.710i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 6.22T + 89T^{2} \)
97 \( 1 + (0.257 + 0.792i)T + (-78.4 + 57.0i)T^{2} \)
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

−11.10426296141263489237568687025, −10.67491610405270073783292122256, −9.547586776383980182083906456892, −8.555671778114868968677945369140, −7.61764681345048024786736361220, −6.07660431964673890113496187062, −5.21629631326852674260199617259, −4.25158821524304183053626283187, −3.38564987644838603532733279728, −0.26718298629666462678012045968, 1.59470394435154743765689264071, 3.40566530257385580711219363772, 4.95227721350620216895642054682, 6.06279672572496198544392325880, 7.00503120662548153972290036566, 7.70421920871280066103733149450, 8.461793631548944650963175391424, 10.21097384720534865901851766635, 11.04366957846045593182646715208, 11.94886573337401307118172179128

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