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Properties

Label 2-322-161.45-c1-0-5
Degree $2$
Conductor $322$
Sign $-0.159 - 0.987i$
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  + (0.5 + 0.866i)2-s + (2.13 + 1.23i)3-s + (−0.499 + 0.866i)4-s + (0.511 + 0.885i)5-s + 2.46i·6-s + (−2.61 − 0.396i)7-s − 0.999·8-s + (1.53 + 2.66i)9-s + (−0.511 + 0.885i)10-s + (1.28 + 0.742i)11-s + (−2.13 + 1.23i)12-s + 0.0815i·13-s + (−0.964 − 2.46i)14-s + 2.52i·15-s + (−0.5 − 0.866i)16-s + (1.20 − 2.08i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (1.23 + 0.711i)3-s + (−0.249 + 0.433i)4-s + (0.228 + 0.395i)5-s + 1.00i·6-s + (−0.988 − 0.149i)7-s − 0.353·8-s + (0.513 + 0.888i)9-s + (−0.161 + 0.279i)10-s + (0.387 + 0.224i)11-s + (−0.616 + 0.355i)12-s + 0.0226i·13-s + (−0.257 − 0.658i)14-s + 0.650i·15-s + (−0.125 − 0.216i)16-s + (0.292 − 0.506i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.159 - 0.987i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.159 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36463 + 1.60307i\)
\(L(\frac12)\) \(\approx\) \(1.36463 + 1.60307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.61 + 0.396i)T \)
23 \( 1 + (-3.64 + 3.11i)T \)
good3 \( 1 + (-2.13 - 1.23i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.511 - 0.885i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.28 - 0.742i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.0815iT - 13T^{2} \)
17 \( 1 + (-1.20 + 2.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.03 - 1.79i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 0.342T + 29T^{2} \)
31 \( 1 + (2.67 + 1.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.135 + 0.0780i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.13iT - 41T^{2} \)
43 \( 1 - 6.69iT - 43T^{2} \)
47 \( 1 + (3.72 - 2.15i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.03 + 4.06i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.48 - 1.43i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.09 - 1.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.5 + 7.84i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.61T + 71T^{2} \)
73 \( 1 + (-5.53 - 3.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.55 + 3.78i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.61T + 83T^{2} \)
89 \( 1 + (-7.27 - 12.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.5T + 97T^{2} \)
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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

−12.12005732126500861101076630724, −10.65746720727275653098738772703, −9.687983240402802412499388941481, −9.178201127826243049607852626108, −8.123952791543737634865472190763, −7.06033213560197283791653825827, −6.14016736996820700841466627441, −4.65068919643822555307878136266, −3.55166782150826415158097849412, −2.73784565340108360657949106252, 1.47184863354212302951381440263, 2.86789035443006757047311345705, 3.66108181233524140662744112991, 5.31985541670502231119058091718, 6.56476866786000613217973308771, 7.59512441305498137722878539925, 8.925072334332176805712594358243, 9.199388533035740438775559232962, 10.35262646501085807297417080649, 11.58405781305097634665569828456

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