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Properties

Label 2-322-161.45-c1-0-8
Degree $2$
Conductor $322$
Sign $0.205 - 0.978i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.205 - 0.978i$
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: \((2,\ 322,\ (\ :1/2),\ 0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71313 + 1.39011i\)
\(L(\frac12)\) \(\approx\) \(1.71313 + 1.39011i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.52 - 1.59i)T \)
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

−11.92843426821637116143736438425, −10.81327838153352598711543689270, −9.723213033333788330631930189007, −8.491287211833345551435578256173, −8.443174399433323534195544869086, −7.28738069775633182560721178387, −5.71839975263887686672710485001, −4.59236219410197615338474971352, −3.82038554419482693477194042280, −2.35974589523856266995407892842, 1.70911562895900039234653316030, 2.72852990613516760878946861298, 3.93096678440454324064202331144, 5.20773932253324997141717068620, 6.81085232668699818698212238132, 7.78603862979739244172844605726, 8.445034199382826436320740720174, 9.532599010506701085724557545045, 10.65285260555443680059256755501, 11.46955389392643430283164659081

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