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Properties

Label 2-322-161.10-c1-0-3
Degree $2$
Conductor $322$
Sign $0.867 - 0.498i$
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 + (−0.409 − 0.795i)3-s + (−0.786 − 0.618i)4-s + (−3.57 + 0.341i)5-s + (0.885 − 0.127i)6-s + (2.38 + 1.15i)7-s + (0.841 − 0.540i)8-s + (1.27 − 1.79i)9-s + (0.846 − 3.48i)10-s + (3.92 − 1.35i)11-s + (−0.169 + 0.878i)12-s + (1.28 + 4.36i)13-s + (−1.86 + 1.87i)14-s + (1.73 + 2.70i)15-s + (0.235 + 0.971i)16-s + (1.39 + 0.558i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.668i)2-s + (−0.236 − 0.458i)3-s + (−0.393 − 0.309i)4-s + (−1.59 + 0.152i)5-s + (0.361 − 0.0519i)6-s + (0.900 + 0.435i)7-s + (0.297 − 0.191i)8-s + (0.425 − 0.597i)9-s + (0.267 − 1.10i)10-s + (1.18 − 0.409i)11-s + (−0.0488 + 0.253i)12-s + (0.355 + 1.21i)13-s + (−0.499 + 0.500i)14-s + (0.448 + 0.697i)15-s + (0.0589 + 0.242i)16-s + (0.338 + 0.135i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.867 - 0.498i$
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: \((2,\ 322,\ (\ :1/2),\ 0.867 - 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.939456 + 0.250694i\)
\(L(\frac12)\) \(\approx\) \(0.939456 + 0.250694i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.38 - 1.15i)T \)
23 \( 1 + (-2.82 - 3.87i)T \)
good3 \( 1 + (0.409 + 0.795i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (3.57 - 0.341i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-3.92 + 1.35i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-1.28 - 4.36i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-1.39 - 0.558i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-2.06 + 0.828i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.850 + 5.91i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-8.60 + 0.409i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (2.64 + 1.88i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-0.727 - 0.332i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-2.28 + 3.55i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (7.95 - 4.59i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.77 - 2.90i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (11.7 + 2.84i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-10.6 - 5.46i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (0.654 + 3.39i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (1.72 + 1.98i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-5.00 + 6.36i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (7.98 - 8.37i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (0.0932 + 0.204i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.0592 - 1.24i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (3.04 - 6.66i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.68328489228022890319370700026, −11.21650145181177198489037975723, −9.508117217745444798534157811837, −8.665568627870274802114227931817, −7.78100755494543603482828147447, −7.00082288654772917670714151207, −6.11925965012471917015897355549, −4.55741143000446571471290525895, −3.72208653788574616161816477543, −1.18745585639654427643624230077, 1.13021196102393849974046537303, 3.36298740042857010510426505457, 4.32524149594173545614808297058, 5.02532793129225311823397840584, 7.06977618284322885416282688278, 7.949069604358224013369339384856, 8.588746157942099512271488966800, 9.984935358633756292227146848189, 10.77570283003876313005376757717, 11.46210334197992456404899735225

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