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Properties

Label 2-351-117.94-c1-0-3
Degree $2$
Conductor $351$
Sign $0.176 - 0.984i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
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  + (0.348 − 0.604i)2-s + (0.756 + 1.31i)4-s + (−1.44 + 2.50i)5-s − 3.17·7-s + 2.45·8-s + (1.00 + 1.74i)10-s + (1.15 − 2.00i)11-s + (0.0625 + 3.60i)13-s + (−1.10 + 1.91i)14-s + (−0.658 + 1.14i)16-s + (−2.69 + 4.66i)17-s + (−2.58 + 4.48i)19-s − 4.37·20-s + (−0.806 − 1.39i)22-s + 6.54·23-s + ⋯
L(s)  = 1  + (0.246 − 0.427i)2-s + (0.378 + 0.655i)4-s + (−0.646 + 1.11i)5-s − 1.19·7-s + 0.866·8-s + (0.318 + 0.552i)10-s + (0.348 − 0.603i)11-s + (0.0173 + 0.999i)13-s + (−0.295 + 0.512i)14-s + (−0.164 + 0.285i)16-s + (−0.653 + 1.13i)17-s + (−0.593 + 1.02i)19-s − 0.978·20-s + (−0.171 − 0.297i)22-s + 1.36·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.176 - 0.984i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ 0.176 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951788 + 0.796364i\)
\(L(\frac12)\) \(\approx\) \(0.951788 + 0.796364i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-0.0625 - 3.60i)T \)
good2 \( 1 + (-0.348 + 0.604i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.44 - 2.50i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.17T + 7T^{2} \)
11 \( 1 + (-1.15 + 2.00i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.69 - 4.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.58 - 4.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.54T + 23T^{2} \)
29 \( 1 + (-2.01 + 3.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.23 + 7.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.42 + 4.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.51T + 41T^{2} \)
43 \( 1 - 5.98T + 43T^{2} \)
47 \( 1 + (0.521 + 0.902i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.29T + 53T^{2} \)
59 \( 1 + (2.35 + 4.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 7.43T + 61T^{2} \)
67 \( 1 - 8.36T + 67T^{2} \)
71 \( 1 + (-0.680 + 1.17i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 + (0.0365 + 0.0633i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.08 - 1.88i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.0891 - 0.154i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.130T + 97T^{2} \)
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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

−11.55155005216089736705111735809, −11.00158343367566913037248124482, −10.13989678576365377064778744887, −8.877634381808336196884795781351, −7.82075989794058372523714408076, −6.73226863805522669654569520209, −6.31606260434636920718593182866, −4.07037600914960685951144671532, −3.52238250337314847652004881653, −2.37856392751040534189125739238, 0.799937844351431221987980337001, 2.91246276970161133771399373761, 4.58773767500309767681368788834, 5.19806332128253861778217015414, 6.63105404519859806939967255533, 7.13649576256637522371004028312, 8.560467073806664387167386274472, 9.356236076436017910442557353810, 10.31902666148822059564909557216, 11.29995987387763199903071424109

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