Properties

Label 2-322-23.16-c1-0-8
Degree $2$
Conductor $322$
Sign $0.347 + 0.937i$
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  + (−0.654 + 0.755i)2-s + (1.39 − 0.894i)3-s + (−0.142 − 0.989i)4-s + (−1.27 − 2.79i)5-s + (−0.235 + 1.63i)6-s + (0.959 − 0.281i)7-s + (0.841 + 0.540i)8-s + (−0.109 + 0.238i)9-s + (2.94 + 0.864i)10-s + (−1.60 − 1.85i)11-s + (−1.08 − 1.25i)12-s + (0.227 + 0.0669i)13-s + (−0.415 + 0.909i)14-s + (−4.27 − 2.74i)15-s + (−0.959 + 0.281i)16-s + (0.786 − 5.46i)17-s + ⋯
L(s)  = 1  + (−0.463 + 0.534i)2-s + (0.803 − 0.516i)3-s + (−0.0711 − 0.494i)4-s + (−0.570 − 1.24i)5-s + (−0.0961 + 0.668i)6-s + (0.362 − 0.106i)7-s + (0.297 + 0.191i)8-s + (−0.0363 + 0.0796i)9-s + (0.931 + 0.273i)10-s + (−0.483 − 0.557i)11-s + (−0.312 − 0.360i)12-s + (0.0632 + 0.0185i)13-s + (−0.111 + 0.243i)14-s + (−1.10 − 0.708i)15-s + (−0.239 + 0.0704i)16-s + (0.190 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.347 + 0.937i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.347 + 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.939455 - 0.653591i\)
\(L(\frac12)\) \(\approx\) \(0.939455 - 0.653591i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (-0.294 + 4.78i)T \)
good3 \( 1 + (-1.39 + 0.894i)T + (1.24 - 2.72i)T^{2} \)
5 \( 1 + (1.27 + 2.79i)T + (-3.27 + 3.77i)T^{2} \)
11 \( 1 + (1.60 + 1.85i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (-0.227 - 0.0669i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.786 + 5.46i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (0.408 + 2.83i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (0.445 - 3.09i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (-4.62 - 2.97i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (1.89 - 4.15i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (-1.02 - 2.24i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-1.86 + 1.19i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + (-4.42 + 1.29i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (7.23 + 2.12i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-10.8 - 6.97i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (9.60 - 11.0i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (3.38 - 3.91i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (1.06 + 7.43i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-3.12 - 0.916i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (5.51 - 12.0i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (-10.5 + 6.74i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-1.83 - 4.01i)T + (-63.5 + 73.3i)T^{2} \)
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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

−11.48646693631075478877705707556, −10.39532886671278560648221694956, −8.969970294101642622476028546096, −8.621696942875175786120772747414, −7.82489807394302292088230436971, −7.00664616133883536305081104808, −5.38539256459372055704560639385, −4.55629847446352813451460179381, −2.71416423244604816773303136407, −0.914720721985671484943307679985, 2.23998149342076785436944675294, 3.38356188929101033984401865869, 4.14996161446858405812705296470, 6.02919161884736636177730987642, 7.44019089562404762449904172172, 8.048498979305104268603743762019, 9.080807735537235872912477785991, 10.11620534275172826905369027998, 10.64689152744132151803770484325, 11.62293656479326339187904182750

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