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Properties

Label 2-322-161.10-c1-0-10
Degree $2$
Conductor $322$
Sign $0.176 + 0.984i$
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  + (−0.327 + 0.945i)2-s + (−1.48 − 2.88i)3-s + (−0.786 − 0.618i)4-s + (1.89 − 0.181i)5-s + (3.21 − 0.461i)6-s + (2.61 + 0.407i)7-s + (0.841 − 0.540i)8-s + (−4.36 + 6.13i)9-s + (−0.449 + 1.85i)10-s + (3.30 − 1.14i)11-s + (−0.614 + 3.18i)12-s + (−1.65 − 5.65i)13-s + (−1.24 + 2.33i)14-s + (−3.34 − 5.20i)15-s + (0.235 + 0.971i)16-s + (−4.27 − 1.71i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.668i)2-s + (−0.858 − 1.66i)3-s + (−0.393 − 0.309i)4-s + (0.849 − 0.0810i)5-s + (1.31 − 0.188i)6-s + (0.988 + 0.154i)7-s + (0.297 − 0.191i)8-s + (−1.45 + 2.04i)9-s + (−0.142 + 0.586i)10-s + (0.995 − 0.344i)11-s + (−0.177 + 0.919i)12-s + (−0.460 − 1.56i)13-s + (−0.331 + 0.624i)14-s + (−0.864 − 1.34i)15-s + (0.0589 + 0.242i)16-s + (−1.03 − 0.415i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.176 + 0.984i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.176 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.770285 - 0.644739i\)
\(L(\frac12)\) \(\approx\) \(0.770285 - 0.644739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.327 - 0.945i)T \)
7 \( 1 + (-2.61 - 0.407i)T \)
23 \( 1 + (4.79 + 0.0501i)T \)
good3 \( 1 + (1.48 + 2.88i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (-1.89 + 0.181i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-3.30 + 1.14i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (1.65 + 5.65i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (4.27 + 1.71i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-3.84 + 1.53i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.382 + 2.66i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-2.61 + 0.124i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (0.522 + 0.372i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-4.39 - 2.00i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (1.69 - 2.64i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (2.04 - 1.18i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.72 - 6.00i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-4.73 - 1.14i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (7.86 + 4.05i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (2.05 + 10.6i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-8.85 - 10.2i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-3.70 + 4.71i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-8.91 + 9.34i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-1.27 - 2.78i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.0232 - 0.487i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (3.79 - 8.32i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.58544013597658389653957705416, −10.67418731439065141581053945638, −9.357864503147090584246604744640, −8.142052975540709367915087623223, −7.57589972320139289042151471346, −6.46018484409975541610560741742, −5.78758155883846732183193428741, −4.97543709273230367676398537905, −2.23878521231595670883340115453, −0.928842297409412973073153932418, 1.89715570214411542991928734652, 3.95116553159342523987711607204, 4.52210157917259771333654349898, 5.60478256013933550002433354734, 6.76265670762218412853161905334, 8.637504726334349143994815806456, 9.464217945321267970293697027074, 9.944382488285661251802350117600, 10.88662247094523809001528176179, 11.63063497656671995727036888117

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