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Properties

Label 2-354-59.26-c1-0-1
Degree $2$
Conductor $354$
Sign $-0.999 - 0.00266i$
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 + (−1.39 + 1.31i)5-s + (−0.161 − 0.986i)6-s + (1.52 + 1.79i)7-s + (0.370 + 0.928i)8-s + (−0.561 − 0.827i)9-s + (0.310 − 1.89i)10-s + (−2.11 + 1.27i)11-s + (0.725 + 0.687i)12-s + (0.133 − 0.197i)13-s + (−2.30 − 0.507i)14-s + (−0.512 − 1.84i)15-s + (−0.856 − 0.515i)16-s + (−0.891 + 1.04i)17-s + ⋯
L(s)  = 1  + (−0.562 + 0.427i)2-s + (−0.270 + 0.510i)3-s + (0.133 − 0.481i)4-s + (−0.622 + 0.589i)5-s + (−0.0660 − 0.402i)6-s + (0.577 + 0.679i)7-s + (0.130 + 0.328i)8-s + (−0.187 − 0.275i)9-s + (0.0980 − 0.598i)10-s + (−0.637 + 0.383i)11-s + (0.209 + 0.198i)12-s + (0.0371 − 0.0547i)13-s + (−0.615 − 0.135i)14-s + (−0.132 − 0.476i)15-s + (−0.214 − 0.128i)16-s + (−0.216 + 0.254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00266i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.999 - 0.00266i$
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: \((2,\ 354,\ (\ :1/2),\ -0.999 - 0.00266i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000672716 + 0.504482i\)
\(L(\frac12)\) \(\approx\) \(0.000672716 + 0.504482i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.40 - 7.55i)T \)
good5 \( 1 + (1.39 - 1.31i)T + (0.270 - 4.99i)T^{2} \)
7 \( 1 + (-1.52 - 1.79i)T + (-1.13 + 6.90i)T^{2} \)
11 \( 1 + (2.11 - 1.27i)T + (5.15 - 9.71i)T^{2} \)
13 \( 1 + (-0.133 + 0.197i)T + (-4.81 - 12.0i)T^{2} \)
17 \( 1 + (0.891 - 1.04i)T + (-2.75 - 16.7i)T^{2} \)
19 \( 1 + (3.44 + 1.59i)T + (12.3 + 14.4i)T^{2} \)
23 \( 1 + (5.80 - 1.95i)T + (18.3 - 13.9i)T^{2} \)
29 \( 1 + (3.70 + 2.81i)T + (7.75 + 27.9i)T^{2} \)
31 \( 1 + (1.25 - 0.580i)T + (20.0 - 23.6i)T^{2} \)
37 \( 1 + (0.0444 - 0.111i)T + (-26.8 - 25.4i)T^{2} \)
41 \( 1 + (0.681 + 0.229i)T + (32.6 + 24.8i)T^{2} \)
43 \( 1 + (-0.00778 - 0.00468i)T + (20.1 + 37.9i)T^{2} \)
47 \( 1 + (-6.61 - 6.26i)T + (2.54 + 46.9i)T^{2} \)
53 \( 1 + (0.243 + 1.48i)T + (-50.2 + 16.9i)T^{2} \)
61 \( 1 + (3.50 - 2.66i)T + (16.3 - 58.7i)T^{2} \)
67 \( 1 + (-0.720 - 1.80i)T + (-48.6 + 46.0i)T^{2} \)
71 \( 1 + (-7.79 - 7.37i)T + (3.84 + 70.8i)T^{2} \)
73 \( 1 + (-2.51 - 0.553i)T + (66.2 + 30.6i)T^{2} \)
79 \( 1 + (-7.51 - 14.1i)T + (-44.3 + 65.3i)T^{2} \)
83 \( 1 + (-13.4 - 1.46i)T + (81.0 + 17.8i)T^{2} \)
89 \( 1 + (-0.118 - 0.0898i)T + (23.8 + 85.7i)T^{2} \)
97 \( 1 + (-10.8 + 2.39i)T + (88.0 - 40.7i)T^{2} \)
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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

−11.58873259956708148313466399283, −10.95452526683714959283630835980, −10.12992009891687236983370086915, −9.103841544085677532228936219327, −8.152813656714077590080080120027, −7.37242824954244858343196665160, −6.16284539263347827484870604704, −5.19269500452914569172578925163, −3.96550243503674923647972325108, −2.28866031072328244995126002172, 0.41478987050724128263211316364, 2.02399717310226885546482416509, 3.80004520320716396623657930408, 4.89052199073188335853559111522, 6.31423367105350803734608681180, 7.61201125394256404503862641102, 8.074332111053743841325878579796, 9.002954202469828828352300806183, 10.37817058213920553174332708094, 10.94836551070666520001943897022

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