Properties

Label 2-354-59.26-c1-0-6
Degree $2$
Conductor $354$
Sign $0.556 + 0.830i$
Analytic cond. $2.82670$
Root an. cond. $1.68128$
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  + (0.796 − 0.605i)2-s + (−0.468 + 0.883i)3-s + (0.267 − 0.963i)4-s + (0.697 − 0.660i)5-s + (0.161 + 0.986i)6-s + (−1.30 − 1.53i)7-s + (−0.370 − 0.928i)8-s + (−0.561 − 0.827i)9-s + (0.155 − 0.947i)10-s + (4.44 − 2.67i)11-s + (0.725 + 0.687i)12-s + (2.29 − 3.37i)13-s + (−1.97 − 0.434i)14-s + (0.256 + 0.925i)15-s + (−0.856 − 0.515i)16-s + (−2.44 + 2.87i)17-s + ⋯
L(s)  = 1  + (0.562 − 0.427i)2-s + (−0.270 + 0.510i)3-s + (0.133 − 0.481i)4-s + (0.311 − 0.295i)5-s + (0.0660 + 0.402i)6-s + (−0.494 − 0.581i)7-s + (−0.130 − 0.328i)8-s + (−0.187 − 0.275i)9-s + (0.0491 − 0.299i)10-s + (1.34 − 0.806i)11-s + (0.209 + 0.198i)12-s + (0.635 − 0.937i)13-s + (−0.527 − 0.116i)14-s + (0.0663 + 0.238i)15-s + (−0.214 − 0.128i)16-s + (−0.593 + 0.698i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.556 + 0.830i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ 0.556 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53079 - 0.816760i\)
\(L(\frac12)\) \(\approx\) \(1.53079 - 0.816760i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.796 + 0.605i)T \)
3 \( 1 + (0.468 - 0.883i)T \)
59 \( 1 + (7.04 - 3.05i)T \)
good5 \( 1 + (-0.697 + 0.660i)T + (0.270 - 4.99i)T^{2} \)
7 \( 1 + (1.30 + 1.53i)T + (-1.13 + 6.90i)T^{2} \)
11 \( 1 + (-4.44 + 2.67i)T + (5.15 - 9.71i)T^{2} \)
13 \( 1 + (-2.29 + 3.37i)T + (-4.81 - 12.0i)T^{2} \)
17 \( 1 + (2.44 - 2.87i)T + (-2.75 - 16.7i)T^{2} \)
19 \( 1 + (-6.22 - 2.87i)T + (12.3 + 14.4i)T^{2} \)
23 \( 1 + (-2.12 + 0.716i)T + (18.3 - 13.9i)T^{2} \)
29 \( 1 + (6.35 + 4.83i)T + (7.75 + 27.9i)T^{2} \)
31 \( 1 + (8.42 - 3.89i)T + (20.0 - 23.6i)T^{2} \)
37 \( 1 + (-1.87 + 4.71i)T + (-26.8 - 25.4i)T^{2} \)
41 \( 1 + (10.1 + 3.43i)T + (32.6 + 24.8i)T^{2} \)
43 \( 1 + (-10.9 - 6.55i)T + (20.1 + 37.9i)T^{2} \)
47 \( 1 + (-4.18 - 3.96i)T + (2.54 + 46.9i)T^{2} \)
53 \( 1 + (-0.601 - 3.66i)T + (-50.2 + 16.9i)T^{2} \)
61 \( 1 + (9.80 - 7.45i)T + (16.3 - 58.7i)T^{2} \)
67 \( 1 + (-4.03 - 10.1i)T + (-48.6 + 46.0i)T^{2} \)
71 \( 1 + (6.41 + 6.07i)T + (3.84 + 70.8i)T^{2} \)
73 \( 1 + (-6.37 - 1.40i)T + (66.2 + 30.6i)T^{2} \)
79 \( 1 + (-5.15 - 9.73i)T + (-44.3 + 65.3i)T^{2} \)
83 \( 1 + (2.12 + 0.231i)T + (81.0 + 17.8i)T^{2} \)
89 \( 1 + (-8.71 - 6.62i)T + (23.8 + 85.7i)T^{2} \)
97 \( 1 + (-0.915 + 0.201i)T + (88.0 - 40.7i)T^{2} \)
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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

−11.17744181855607104207033246358, −10.67389649665929524218004916976, −9.535625270156219383668722153529, −8.927120747043069749614111974576, −7.38285395893995495018708189113, −6.07284876163421763006101025090, −5.51833784182615249997091356545, −3.96532466563473445355991313283, −3.39855960965552789897758422843, −1.21481904543481366279174137604, 1.94359247921457287421211396065, 3.45108614093424650789323183586, 4.80071373596143556038219146316, 5.96634716010650091950263263003, 6.77690100182168212999491269276, 7.35545110931400123725986970910, 9.065619896073751430991957105688, 9.383659629179231621587934270286, 11.07710256415695482193518227366, 11.77335336228340876106452648708

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