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Properties

Label 2-322-161.19-c1-0-6
Degree $2$
Conductor $322$
Sign $0.398 + 0.917i$
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  + (−0.995 − 0.0950i)2-s + (−0.855 + 0.897i)3-s + (0.981 + 0.189i)4-s + (−2.48 + 1.28i)5-s + (0.937 − 0.811i)6-s + (−1.18 − 2.36i)7-s + (−0.959 − 0.281i)8-s + (0.0696 + 1.46i)9-s + (2.60 − 1.04i)10-s + (−0.550 − 5.76i)11-s + (−1.00 + 0.719i)12-s + (3.86 + 0.556i)13-s + (0.951 + 2.46i)14-s + (0.978 − 3.33i)15-s + (0.928 + 0.371i)16-s + (1.36 − 3.94i)17-s + ⋯
L(s)  = 1  + (−0.703 − 0.0672i)2-s + (−0.493 + 0.518i)3-s + (0.490 + 0.0946i)4-s + (−1.11 + 0.574i)5-s + (0.382 − 0.331i)6-s + (−0.446 − 0.894i)7-s + (−0.339 − 0.0996i)8-s + (0.0232 + 0.487i)9-s + (0.822 − 0.329i)10-s + (−0.165 − 1.73i)11-s + (−0.291 + 0.207i)12-s + (1.07 + 0.154i)13-s + (0.254 + 0.659i)14-s + (0.252 − 0.860i)15-s + (0.232 + 0.0929i)16-s + (0.330 − 0.956i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.398 + 0.917i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.398 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.398139 - 0.261212i\)
\(L(\frac12)\) \(\approx\) \(0.398139 - 0.261212i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.995 + 0.0950i)T \)
7 \( 1 + (1.18 + 2.36i)T \)
23 \( 1 + (-3.73 + 3.01i)T \)
good3 \( 1 + (0.855 - 0.897i)T + (-0.142 - 2.99i)T^{2} \)
5 \( 1 + (2.48 - 1.28i)T + (2.90 - 4.07i)T^{2} \)
11 \( 1 + (0.550 + 5.76i)T + (-10.8 + 2.08i)T^{2} \)
13 \( 1 + (-3.86 - 0.556i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (-1.36 + 3.94i)T + (-13.3 - 10.5i)T^{2} \)
19 \( 1 + (-1.17 - 3.40i)T + (-14.9 + 11.7i)T^{2} \)
29 \( 1 + (5.50 + 6.35i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-7.23 + 1.75i)T + (27.5 - 14.2i)T^{2} \)
37 \( 1 + (2.02 - 0.0962i)T + (36.8 - 3.51i)T^{2} \)
41 \( 1 + (-0.229 + 0.357i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (3.34 + 11.4i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (3.62 + 2.09i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.28 + 5.44i)T + (-12.4 + 51.5i)T^{2} \)
59 \( 1 + (-2.01 - 5.02i)T + (-42.7 + 40.7i)T^{2} \)
61 \( 1 + (0.249 - 0.238i)T + (2.90 - 60.9i)T^{2} \)
67 \( 1 + (2.68 + 1.91i)T + (21.9 + 63.3i)T^{2} \)
71 \( 1 + (-5.67 - 12.4i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-0.349 + 1.81i)T + (-67.7 - 27.1i)T^{2} \)
79 \( 1 + (-7.07 + 8.99i)T + (-18.6 - 76.7i)T^{2} \)
83 \( 1 + (8.99 - 5.78i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (2.74 - 11.2i)T + (-79.1 - 40.7i)T^{2} \)
97 \( 1 + (0.578 + 0.371i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.21458002777214321984493539774, −10.69711746668266133173527297941, −9.839778616175031004097823438952, −8.465633782667978379117890990206, −7.79071774009742899328799279963, −6.75364414550323452006970066958, −5.64271524311222633191252699239, −4.00117088677496626553997360463, −3.18547483724077247614703903396, −0.49756553523500331460463944916, 1.40999125800022105775773803354, 3.37236160419274127271571026140, 4.88679852568286153155942574058, 6.18573993107214402288054586896, 7.07748029528445306552293330841, 8.013888809849056559572367341808, 8.954828392159580236227675759136, 9.730321381078417606249867946879, 11.07123349150589409964873543209, 11.81411883152936648411817305576

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