Properties

Label 2-322-23.8-c1-0-8
Degree $2$
Conductor $322$
Sign $-0.639 + 0.768i$
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  + (−0.142 + 0.989i)2-s + (−1.08 − 2.38i)3-s + (−0.959 − 0.281i)4-s + (0.443 + 0.512i)5-s + (2.51 − 0.739i)6-s + (−0.841 − 0.540i)7-s + (0.415 − 0.909i)8-s + (−2.54 + 2.93i)9-s + (−0.569 + 0.366i)10-s + (−0.529 − 3.68i)11-s + (0.373 + 2.59i)12-s + (0.0806 − 0.0518i)13-s + (0.654 − 0.755i)14-s + (0.738 − 1.61i)15-s + (0.841 + 0.540i)16-s + (−7.52 + 2.20i)17-s + ⋯
L(s)  = 1  + (−0.100 + 0.699i)2-s + (−0.629 − 1.37i)3-s + (−0.479 − 0.140i)4-s + (0.198 + 0.228i)5-s + (1.02 − 0.301i)6-s + (−0.317 − 0.204i)7-s + (0.146 − 0.321i)8-s + (−0.847 + 0.977i)9-s + (−0.180 + 0.115i)10-s + (−0.159 − 1.11i)11-s + (0.107 + 0.749i)12-s + (0.0223 − 0.0143i)13-s + (0.175 − 0.201i)14-s + (0.190 − 0.417i)15-s + (0.210 + 0.135i)16-s + (−1.82 + 0.535i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.639 + 0.768i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.639 + 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228722 - 0.488093i\)
\(L(\frac12)\) \(\approx\) \(0.228722 - 0.488093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.142 - 0.989i)T \)
7 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (-4.07 + 2.52i)T \)
good3 \( 1 + (1.08 + 2.38i)T + (-1.96 + 2.26i)T^{2} \)
5 \( 1 + (-0.443 - 0.512i)T + (-0.711 + 4.94i)T^{2} \)
11 \( 1 + (0.529 + 3.68i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (-0.0806 + 0.0518i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (7.52 - 2.20i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (7.32 + 2.15i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (3.73 - 1.09i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (0.242 - 0.530i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (-4.26 + 4.92i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (-3.00 - 3.47i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-0.547 - 1.19i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 + (-4.71 - 3.03i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (0.235 - 0.151i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (-2.64 + 5.79i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (-2.21 + 15.4i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (0.800 - 5.56i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (4.11 + 1.20i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (12.7 - 8.20i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-10.1 + 11.7i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-0.279 - 0.611i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (4.10 + 4.74i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.09694462917123346035268058014, −10.74595267900737405512375984271, −9.018202425301303128573663587655, −8.316570434329833695948353517096, −7.13187175027520800983312190711, −6.43549494686813108021038822844, −5.91643142653035493585474921630, −4.35361678165936203274489874846, −2.34808446162304431745323474446, −0.40959955859834572998435046716, 2.31752316498821086293315156977, 4.00575522265361037204397062138, 4.66053801709701546677402144874, 5.70673249444634605184108307766, 7.07287337141798412445088760149, 8.810344427451996648772541512622, 9.349258258310398531293948257479, 10.21264227832306153134164637377, 10.92450669157320912635741139440, 11.63775264788850873214978232208

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