Properties

Label 2-322-23.18-c1-0-9
Degree $2$
Conductor $322$
Sign $-0.226 + 0.974i$
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  + (−0.959 − 0.281i)2-s + (1.19 − 1.37i)3-s + (0.841 + 0.540i)4-s + (0.248 − 1.73i)5-s + (−1.52 + 0.982i)6-s + (−0.415 − 0.909i)7-s + (−0.654 − 0.755i)8-s + (−0.0432 − 0.300i)9-s + (−0.726 + 1.59i)10-s + (0.896 − 0.263i)11-s + (1.74 − 0.512i)12-s + (2.78 − 6.09i)13-s + (0.142 + 0.989i)14-s + (−2.08 − 2.40i)15-s + (0.415 + 0.909i)16-s + (−6.32 + 4.06i)17-s + ⋯
L(s)  = 1  + (−0.678 − 0.199i)2-s + (0.687 − 0.793i)3-s + (0.420 + 0.270i)4-s + (0.111 − 0.774i)5-s + (−0.624 + 0.401i)6-s + (−0.157 − 0.343i)7-s + (−0.231 − 0.267i)8-s + (−0.0144 − 0.100i)9-s + (−0.229 + 0.503i)10-s + (0.270 − 0.0793i)11-s + (0.503 − 0.147i)12-s + (0.771 − 1.68i)13-s + (0.0380 + 0.264i)14-s + (−0.537 − 0.620i)15-s + (0.103 + 0.227i)16-s + (−1.53 + 0.985i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.226 + 0.974i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.226 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.734209 - 0.924322i\)
\(L(\frac12)\) \(\approx\) \(0.734209 - 0.924322i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (-4.64 - 1.19i)T \)
good3 \( 1 + (-1.19 + 1.37i)T + (-0.426 - 2.96i)T^{2} \)
5 \( 1 + (-0.248 + 1.73i)T + (-4.79 - 1.40i)T^{2} \)
11 \( 1 + (-0.896 + 0.263i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-2.78 + 6.09i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (6.32 - 4.06i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (1.74 + 1.11i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (2.21 - 1.42i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (6.49 + 7.50i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-0.435 - 3.03i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (-1.15 + 8.00i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-3.49 + 4.03i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + 1.68T + 47T^{2} \)
53 \( 1 + (-5.55 - 12.1i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (3.18 - 6.97i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (-4.71 - 5.43i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (-9.43 - 2.77i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (-7.07 - 2.07i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-10.8 - 6.94i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (0.418 - 0.916i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (1.27 + 8.84i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (11.4 - 13.1i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (1.61 - 11.1i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.05976930847401280977866494365, −10.57122663717611028522666475706, −9.076848537955813189146812516334, −8.631441389234655877932746640694, −7.76290757964701407005884990228, −6.82936848629088816921590567653, −5.54490259370748172919919549924, −3.87670329589017840395382654798, −2.42063148783579566623642166621, −1.04449043167929019259401788373, 2.19547716228021400069805984596, 3.47722921642058864170596283105, 4.70946005637435790967694491013, 6.54449865400818470166853017820, 6.86128493893030461897746258749, 8.543094193975477072836483946369, 9.131519809735793257403217838885, 9.691635382430924644240203015373, 10.99172672169760696887089055153, 11.33755967080857797768819355655

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