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Properties

Label 2-322-161.66-c1-0-2
Degree $2$
Conductor $322$
Sign $-0.755 - 0.655i$
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  + (−0.786 + 0.618i)2-s + (0.118 − 0.0843i)3-s + (0.235 − 0.971i)4-s + (−0.327 − 0.0631i)5-s + (−0.0409 + 0.139i)6-s + (−2.63 + 0.186i)7-s + (0.415 + 0.909i)8-s + (−0.974 + 2.81i)9-s + (0.296 − 0.152i)10-s + (−1.10 + 1.39i)11-s + (−0.0540 − 0.134i)12-s + (−0.177 + 0.275i)13-s + (1.95 − 1.77i)14-s + (−0.0441 + 0.0201i)15-s + (−0.888 − 0.458i)16-s + (−4.15 + 3.96i)17-s + ⋯
L(s)  = 1  + (−0.555 + 0.437i)2-s + (0.0683 − 0.0486i)3-s + (0.117 − 0.485i)4-s + (−0.146 − 0.0282i)5-s + (−0.0167 + 0.0569i)6-s + (−0.997 + 0.0703i)7-s + (0.146 + 0.321i)8-s + (−0.324 + 0.938i)9-s + (0.0938 − 0.0483i)10-s + (−0.331 + 0.421i)11-s + (−0.0155 − 0.0389i)12-s + (−0.0491 + 0.0764i)13-s + (0.523 − 0.475i)14-s + (−0.0113 + 0.00520i)15-s + (−0.222 − 0.114i)16-s + (−1.00 + 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.755 - 0.655i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.755 - 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.191316 + 0.512682i\)
\(L(\frac12)\) \(\approx\) \(0.191316 + 0.512682i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.786 - 0.618i)T \)
7 \( 1 + (2.63 - 0.186i)T \)
23 \( 1 + (-3.70 - 3.04i)T \)
good3 \( 1 + (-0.118 + 0.0843i)T + (0.981 - 2.83i)T^{2} \)
5 \( 1 + (0.327 + 0.0631i)T + (4.64 + 1.85i)T^{2} \)
11 \( 1 + (1.10 - 1.39i)T + (-2.59 - 10.6i)T^{2} \)
13 \( 1 + (0.177 - 0.275i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (4.15 - 3.96i)T + (0.808 - 16.9i)T^{2} \)
19 \( 1 + (-5.96 - 5.69i)T + (0.904 + 18.9i)T^{2} \)
29 \( 1 + (7.55 + 2.21i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-0.379 + 3.97i)T + (-30.4 - 5.86i)T^{2} \)
37 \( 1 + (4.83 + 1.67i)T + (29.0 + 22.8i)T^{2} \)
41 \( 1 + (-3.15 - 2.72i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (1.38 + 0.632i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + (8.10 - 4.68i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.3 + 0.539i)T + (52.7 - 5.03i)T^{2} \)
59 \( 1 + (-0.518 - 1.00i)T + (-34.2 + 48.0i)T^{2} \)
61 \( 1 + (-2.89 + 4.07i)T + (-19.9 - 57.6i)T^{2} \)
67 \( 1 + (-3.96 + 9.90i)T + (-48.4 - 46.2i)T^{2} \)
71 \( 1 + (-0.802 - 5.57i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (13.6 + 3.30i)T + (64.8 + 33.4i)T^{2} \)
79 \( 1 + (-12.3 - 0.587i)T + (78.6 + 7.50i)T^{2} \)
83 \( 1 + (-11.3 - 13.1i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (-14.5 + 1.39i)T + (87.3 - 16.8i)T^{2} \)
97 \( 1 + (-1.49 + 1.72i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.86693066503435922785505656714, −10.87931269510360348873767883496, −9.956977213717254186782548383833, −9.242777348966371804894272879847, −8.048089813796639339993277518132, −7.40991716620018311692727223019, −6.19031431043324812689631980910, −5.29488355601525627783100032322, −3.71128299781187602070101473278, −2.08108689811349439609384470076, 0.44305716328781292517841219139, 2.76652236432615449661216301952, 3.60270653380076684284408079768, 5.25838214371584218643134468720, 6.67547330337363718118858419222, 7.34591712636675578710863949510, 8.937249626451241040949695467846, 9.197121081757494095625055471558, 10.26124780669403580390910163430, 11.34458299163805917129532838321

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