Properties

Label 2-322-161.32-c1-0-13
Degree $2$
Conductor $322$
Sign $0.895 - 0.444i$
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  + (0.723 + 0.690i)2-s + (1.57 − 0.304i)3-s + (0.0475 + 0.998i)4-s + (1.18 − 0.928i)5-s + (1.35 + 0.869i)6-s + (2.64 − 0.0960i)7-s + (−0.654 + 0.755i)8-s + (−0.382 + 0.152i)9-s + (1.49 + 0.142i)10-s + (−3.07 + 2.92i)11-s + (0.379 + 1.56i)12-s + (−2.57 − 5.63i)13-s + (1.97 + 1.75i)14-s + (1.58 − 1.82i)15-s + (−0.995 + 0.0950i)16-s + (1.27 − 0.657i)17-s + ⋯
L(s)  = 1  + (0.511 + 0.487i)2-s + (0.912 − 0.175i)3-s + (0.0237 + 0.499i)4-s + (0.528 − 0.415i)5-s + (0.552 + 0.355i)6-s + (0.999 − 0.0363i)7-s + (−0.231 + 0.267i)8-s + (−0.127 + 0.0509i)9-s + (0.473 + 0.0451i)10-s + (−0.926 + 0.883i)11-s + (0.109 + 0.451i)12-s + (−0.713 − 1.56i)13-s + (0.529 + 0.469i)14-s + (0.408 − 0.471i)15-s + (−0.248 + 0.0237i)16-s + (0.309 − 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29242 + 0.537495i\)
\(L(\frac12)\) \(\approx\) \(2.29242 + 0.537495i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.723 - 0.690i)T \)
7 \( 1 + (-2.64 + 0.0960i)T \)
23 \( 1 + (-4.33 - 2.04i)T \)
good3 \( 1 + (-1.57 + 0.304i)T + (2.78 - 1.11i)T^{2} \)
5 \( 1 + (-1.18 + 0.928i)T + (1.17 - 4.85i)T^{2} \)
11 \( 1 + (3.07 - 2.92i)T + (0.523 - 10.9i)T^{2} \)
13 \( 1 + (2.57 + 5.63i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-1.27 + 0.657i)T + (9.86 - 13.8i)T^{2} \)
19 \( 1 + (1.55 + 0.803i)T + (11.0 + 15.4i)T^{2} \)
29 \( 1 + (-1.35 - 0.868i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-0.231 - 0.668i)T + (-24.3 + 19.1i)T^{2} \)
37 \( 1 + (5.54 - 2.21i)T + (26.7 - 25.5i)T^{2} \)
41 \( 1 + (-0.620 - 4.31i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (6.27 + 7.23i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + (2.01 + 3.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.18 + 10.0i)T + (-17.3 + 50.0i)T^{2} \)
59 \( 1 + (-8.27 - 0.789i)T + (57.9 + 11.1i)T^{2} \)
61 \( 1 + (-0.534 - 0.102i)T + (56.6 + 22.6i)T^{2} \)
67 \( 1 + (1.92 - 7.91i)T + (-59.5 - 30.7i)T^{2} \)
71 \( 1 + (-13.2 + 3.90i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (0.243 + 5.11i)T + (-72.6 + 6.93i)T^{2} \)
79 \( 1 + (-3.18 + 4.47i)T + (-25.8 - 74.6i)T^{2} \)
83 \( 1 + (-1.17 + 8.17i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (4.82 - 13.9i)T + (-69.9 - 55.0i)T^{2} \)
97 \( 1 + (1.87 + 13.0i)T + (-93.0 + 27.3i)T^{2} \)
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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

−11.95232245257902800224804003146, −10.70596635689420463118803107377, −9.709339181957661589067380095229, −8.521718155780520979687849568877, −7.920113466094102747880942188617, −7.13519187431749470695242984441, −5.22853197056115438844539487605, −5.13634975058019340690319401181, −3.22710504534102406418712236829, −2.08214496751002614210797612376, 2.03740817456931949844461498072, 2.92709805040134232442284672023, 4.29564880262309534888815321598, 5.39599164990234834612081093146, 6.59745258872926404530463824848, 7.958477675092943679200068142260, 8.800164714757902966710579796364, 9.737814965305498356068893928035, 10.75864307373984890405940567383, 11.45483496294141305051565850361

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