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Properties

Label 2-322-161.10-c1-0-6
Degree $2$
Conductor $322$
Sign $0.966 - 0.257i$
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 + (−0.400 − 0.777i)3-s + (−0.786 − 0.618i)4-s + (2.83 − 0.270i)5-s + (0.865 − 0.124i)6-s + (−1.59 + 2.11i)7-s + (0.841 − 0.540i)8-s + (1.29 − 1.82i)9-s + (−0.671 + 2.76i)10-s + (1.00 − 0.347i)11-s + (−0.165 + 0.858i)12-s + (−0.757 − 2.58i)13-s + (−1.47 − 2.19i)14-s + (−1.34 − 2.09i)15-s + (0.235 + 0.971i)16-s + (6.94 + 2.78i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.668i)2-s + (−0.231 − 0.448i)3-s + (−0.393 − 0.309i)4-s + (1.26 − 0.121i)5-s + (0.353 − 0.0508i)6-s + (−0.603 + 0.797i)7-s + (0.297 − 0.191i)8-s + (0.432 − 0.606i)9-s + (−0.212 + 0.875i)10-s + (0.302 − 0.104i)11-s + (−0.0477 + 0.247i)12-s + (−0.210 − 0.715i)13-s + (−0.393 − 0.587i)14-s + (−0.347 − 0.541i)15-s + (0.0589 + 0.242i)16-s + (1.68 + 0.674i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.966 - 0.257i$
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: \((2,\ 322,\ (\ :1/2),\ 0.966 - 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28037 + 0.167586i\)
\(L(\frac12)\) \(\approx\) \(1.28037 + 0.167586i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 2.11i)T \)
23 \( 1 + (-4.04 + 2.57i)T \)
good3 \( 1 + (0.400 + 0.777i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (-2.83 + 0.270i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-1.00 + 0.347i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (0.757 + 2.58i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-6.94 - 2.78i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (0.186 - 0.0745i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.433 - 3.01i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-4.29 + 0.204i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-5.57 - 3.96i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (6.77 + 3.09i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (0.434 - 0.676i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (8.79 - 5.07i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.541 + 0.568i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (7.31 + 1.77i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (9.54 + 4.92i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-2.07 - 10.7i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (5.78 + 6.67i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-0.744 + 0.946i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (1.77 - 1.86i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (2.34 + 5.13i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.601 + 12.6i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-3.76 + 8.25i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.94343847533876109630031376839, −10.29796436111209256591296409107, −9.741979560634859602736189334782, −8.924613953610178440931320942154, −7.79932797316695968751895039780, −6.48868801160687200334465785493, −6.05387646090051867563522703034, −5.11195029531251185892934487114, −3.15074130232949430893063286540, −1.34990714247198665007787700080, 1.50883336965410399533229099805, 3.04598555068810450445788309168, 4.40507783032116625658814282862, 5.48628189504625192978954986484, 6.74187852235545557062932502284, 7.78556980673736397522656399429, 9.446086993904534783443548601066, 9.759283063677691049198604325860, 10.41696171635749222740971115499, 11.39865317129410513844580408703

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