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Properties

Label 2-322-161.111-c1-0-7
Degree $2$
Conductor $322$
Sign $0.968 - 0.250i$
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 + (−0.192 − 0.656i)3-s + (−0.654 + 0.755i)4-s + (−2.27 − 1.46i)5-s + (0.516 − 0.447i)6-s + (2.42 + 1.04i)7-s + (−0.959 − 0.281i)8-s + (2.13 − 1.36i)9-s + (0.385 − 2.68i)10-s + (2.26 + 1.03i)11-s + (0.622 + 0.284i)12-s + (6.52 + 0.938i)13-s + (0.0563 + 2.64i)14-s + (−0.521 + 1.77i)15-s + (−0.142 − 0.989i)16-s + (−0.316 − 0.365i)17-s + ⋯
L(s)  = 1  + (0.293 + 0.643i)2-s + (−0.111 − 0.378i)3-s + (−0.327 + 0.377i)4-s + (−1.01 − 0.655i)5-s + (0.210 − 0.182i)6-s + (0.918 + 0.395i)7-s + (−0.339 − 0.0996i)8-s + (0.710 − 0.456i)9-s + (0.121 − 0.848i)10-s + (0.681 + 0.311i)11-s + (0.179 + 0.0820i)12-s + (1.81 + 0.260i)13-s + (0.0150 + 0.706i)14-s + (−0.134 + 0.459i)15-s + (−0.0355 − 0.247i)16-s + (−0.0767 − 0.0885i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45295 + 0.184735i\)
\(L(\frac12)\) \(\approx\) \(1.45295 + 0.184735i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.42 - 1.04i)T \)
23 \( 1 + (4.68 - 1.03i)T \)
good3 \( 1 + (0.192 + 0.656i)T + (-2.52 + 1.62i)T^{2} \)
5 \( 1 + (2.27 + 1.46i)T + (2.07 + 4.54i)T^{2} \)
11 \( 1 + (-2.26 - 1.03i)T + (7.20 + 8.31i)T^{2} \)
13 \( 1 + (-6.52 - 0.938i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (0.316 + 0.365i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-2.14 + 2.47i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (0.302 + 0.349i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-0.191 + 0.651i)T + (-26.0 - 16.7i)T^{2} \)
37 \( 1 + (5.71 + 8.89i)T + (-15.3 + 33.6i)T^{2} \)
41 \( 1 + (4.71 - 7.33i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (-2.55 - 8.69i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 - 2.60iT - 47T^{2} \)
53 \( 1 + (1.06 - 0.152i)T + (50.8 - 14.9i)T^{2} \)
59 \( 1 + (-10.1 - 1.46i)T + (56.6 + 16.6i)T^{2} \)
61 \( 1 + (4.83 + 1.41i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (2.73 - 1.24i)T + (43.8 - 50.6i)T^{2} \)
71 \( 1 + (-0.585 - 1.28i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-1.75 - 1.52i)T + (10.3 + 72.2i)T^{2} \)
79 \( 1 + (-2.36 - 0.340i)T + (75.7 + 22.2i)T^{2} \)
83 \( 1 + (-2.53 + 1.62i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (13.2 - 3.87i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (6.98 + 4.49i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.80141087213172002320717308662, −11.15516296500688516773462402521, −9.446739598462861480105637809164, −8.551939287472780039195479392302, −7.87346832864983731691240654858, −6.86144318763710548336676650759, −5.82220226556485544838448648548, −4.48376224441225020852597255081, −3.81992224724297796585777737419, −1.33763018950170019157811588651, 1.51247839195738327349801581535, 3.65092196870431179799329813450, 3.99955910988497251811599243944, 5.37785949568370030347045263576, 6.75538003913942922558766016800, 7.911299251416704985020665126586, 8.696597279145267788956231379306, 10.23244985465621026440009254134, 10.75292198781468184248551754340, 11.48424615078051058388864718833

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