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Properties

Label 2-322-161.45-c1-0-14
Degree $2$
Conductor $322$
Sign $-0.854 + 0.519i$
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  + (0.5 + 0.866i)2-s + (−1.07 − 0.622i)3-s + (−0.499 + 0.866i)4-s + (−0.668 − 1.15i)5-s − 1.24i·6-s + (−2.09 + 1.61i)7-s − 0.999·8-s + (−0.726 − 1.25i)9-s + (0.668 − 1.15i)10-s + (−5.22 − 3.01i)11-s + (1.07 − 0.622i)12-s + 6.52i·13-s + (−2.44 − 1.00i)14-s + 1.66i·15-s + (−0.5 − 0.866i)16-s + (−0.568 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.622 − 0.359i)3-s + (−0.249 + 0.433i)4-s + (−0.299 − 0.518i)5-s − 0.507i·6-s + (−0.791 + 0.610i)7-s − 0.353·8-s + (−0.242 − 0.419i)9-s + (0.211 − 0.366i)10-s + (−1.57 − 0.909i)11-s + (0.311 − 0.179i)12-s + 1.80i·13-s + (−0.653 − 0.269i)14-s + 0.429i·15-s + (−0.125 − 0.216i)16-s + (−0.137 + 0.238i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.854 + 0.519i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.854 + 0.519i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0159075 - 0.0567342i\)
\(L(\frac12)\) \(\approx\) \(0.0159075 - 0.0567342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.09 - 1.61i)T \)
23 \( 1 + (1.35 + 4.59i)T \)
good3 \( 1 + (1.07 + 0.622i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.668 + 1.15i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.22 + 3.01i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.52iT - 13T^{2} \)
17 \( 1 + (0.568 - 0.984i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.77 + 3.07i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 3.18T + 29T^{2} \)
31 \( 1 + (-2.80 - 1.61i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.10 - 2.94i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.928iT - 41T^{2} \)
43 \( 1 + 0.303iT - 43T^{2} \)
47 \( 1 + (0.243 - 0.140i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.3 - 6.56i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.10 + 0.635i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.00 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.57 + 2.06i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.02T + 71T^{2} \)
73 \( 1 + (11.1 + 6.40i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.6 - 6.13i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.99T + 83T^{2} \)
89 \( 1 + (4.71 + 8.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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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

−11.58514055786839728667724182243, −10.41191380601900519804460217481, −8.919318014201427669917519475506, −8.568128507949488990774441448453, −7.06335548560126162846349860439, −6.29373181934555341942116393356, −5.46027204891107236025542600232, −4.29800786812570361700191756820, −2.75062683859895025633003967121, −0.03748518596492980902858070773, 2.60628709238562299731206157607, 3.67821495906757859919396709099, 5.08467736981552967891785917407, 5.75096572058636227705360198619, 7.26052467185504412566814130368, 8.057286656061386007340726090704, 9.832095011808870847606288443594, 10.41669762110185673632389061015, 10.80512695685898586069122698938, 11.93964580860079001427594358596

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