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Properties

Label 2-352-88.51-c1-0-5
Degree $2$
Conductor $352$
Sign $0.966 + 0.254i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.625 − 1.92i)3-s + (1.75 + 2.41i)5-s + (0.216 + 0.667i)7-s + (−0.888 − 0.645i)9-s + (2.75 + 1.85i)11-s + (2.50 + 1.82i)13-s + (5.73 − 1.86i)15-s + (−2.54 − 3.49i)17-s + (−4.99 − 1.62i)19-s + 1.42·21-s − 0.883i·23-s + (−1.19 + 3.68i)25-s + (3.11 − 2.26i)27-s + (−3.13 − 9.66i)29-s + (−4.71 + 6.48i)31-s + ⋯
L(s)  = 1  + (0.361 − 1.11i)3-s + (0.783 + 1.07i)5-s + (0.0819 + 0.252i)7-s + (−0.296 − 0.215i)9-s + (0.829 + 0.558i)11-s + (0.695 + 0.505i)13-s + (1.48 − 0.481i)15-s + (−0.616 − 0.848i)17-s + (−1.14 − 0.372i)19-s + 0.310·21-s − 0.184i·23-s + (−0.239 + 0.737i)25-s + (0.599 − 0.435i)27-s + (−0.583 − 1.79i)29-s + (−0.846 + 1.16i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.966 + 0.254i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.966 + 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73755 - 0.225137i\)
\(L(\frac12)\) \(\approx\) \(1.73755 - 0.225137i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.75 - 1.85i)T \)
good3 \( 1 + (-0.625 + 1.92i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.75 - 2.41i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.216 - 0.667i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.50 - 1.82i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.54 + 3.49i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.99 + 1.62i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.883iT - 23T^{2} \)
29 \( 1 + (3.13 + 9.66i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.71 - 6.48i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.84 + 0.924i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.79 - 0.906i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.57iT - 43T^{2} \)
47 \( 1 + (4.57 + 1.48i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.22 - 9.94i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.63 - 5.02i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.21 - 4.51i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 5.11T + 67T^{2} \)
71 \( 1 + (4.66 + 6.42i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.05 - 1.31i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.28 - 1.65i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.679 - 0.934i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 9.59T + 89T^{2} \)
97 \( 1 + (-5.12 - 3.72i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.46420773750072091852846420065, −10.61392512375839290457841242160, −9.488867630987447717537945479207, −8.660225747443238524536072589089, −7.37643915467477048647960522106, −6.69977627972717519369523907722, −6.06845546696229875963809803602, −4.29992759771187308770980752143, −2.61896264132192363270788793994, −1.79848368273680931550693282686, 1.55570748409037766801217136484, 3.57187090523784214348679985546, 4.37386998652082777272479551527, 5.50972315383922853435973378770, 6.45907948155804718256140043234, 8.203791195402211713557395344469, 8.961349743521170157624431041098, 9.465329189986587773793941456075, 10.54989310477775856610883859033, 11.18100502914202855208399116357

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