Properties

Label 2-322-23.4-c1-0-1
Degree $2$
Conductor $322$
Sign $-0.858 - 0.512i$
Analytic cond. $2.57118$
Root an. cond. $1.60349$
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  + (0.415 + 0.909i)2-s + (−0.195 + 0.0575i)3-s + (−0.654 + 0.755i)4-s + (−1.56 − 1.00i)5-s + (−0.133 − 0.154i)6-s + (0.142 + 0.989i)7-s + (−0.959 − 0.281i)8-s + (−2.48 + 1.59i)9-s + (0.265 − 1.84i)10-s + (−2.33 + 5.11i)11-s + (0.0848 − 0.185i)12-s + (−0.665 + 4.62i)13-s + (−0.841 + 0.540i)14-s + (0.364 + 0.107i)15-s + (−0.142 − 0.989i)16-s + (2.86 + 3.30i)17-s + ⋯
L(s)  = 1  + (0.293 + 0.643i)2-s + (−0.113 + 0.0332i)3-s + (−0.327 + 0.377i)4-s + (−0.700 − 0.450i)5-s + (−0.0545 − 0.0629i)6-s + (0.0537 + 0.374i)7-s + (−0.339 − 0.0996i)8-s + (−0.829 + 0.533i)9-s + (0.0838 − 0.582i)10-s + (−0.703 + 1.54i)11-s + (0.0244 − 0.0536i)12-s + (−0.184 + 1.28i)13-s + (−0.224 + 0.144i)14-s + (0.0941 + 0.0276i)15-s + (−0.0355 − 0.247i)16-s + (0.695 + 0.802i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 - 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.858 - 0.512i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.858 - 0.512i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.230594 + 0.836545i\)
\(L(\frac12)\) \(\approx\) \(0.230594 + 0.836545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (4.17 + 2.35i)T \)
good3 \( 1 + (0.195 - 0.0575i)T + (2.52 - 1.62i)T^{2} \)
5 \( 1 + (1.56 + 1.00i)T + (2.07 + 4.54i)T^{2} \)
11 \( 1 + (2.33 - 5.11i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.665 - 4.62i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-2.86 - 3.30i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-3.79 + 4.37i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (0.269 + 0.311i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-0.252 - 0.0741i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-7.85 + 5.05i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-4.44 - 2.85i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (6.06 - 1.77i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 - 0.945T + 47T^{2} \)
53 \( 1 + (-0.751 - 5.22i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (1.34 - 9.38i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (4.55 + 1.33i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-5.30 - 11.6i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (3.10 + 6.80i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-7.28 + 8.40i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (1.93 - 13.4i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (9.76 - 6.27i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-4.75 + 1.39i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-14.1 - 9.11i)T + (40.2 + 88.2i)T^{2} \)
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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

−12.09295731783080235569603594665, −11.37261024064473688750032061143, −10.01356857467679414690725292272, −9.025954148250233941118394677712, −8.013410643051995043501104240201, −7.37406981867544818823006879641, −6.07911580568653930137097076258, −4.94524324948006678320505923913, −4.23718168879569642935662094604, −2.42957670686098912782228233981, 0.56502559059599166701912485776, 3.10478317924594206825259204531, 3.47777943494357398106885198996, 5.31450793667514637430551828366, 5.99527562588540511282377343815, 7.67321554104047503998850802066, 8.223762440865518515053659579617, 9.647057903370879754501290918174, 10.52323723840281901688384727181, 11.41709740666879009328259113670

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