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Properties

Label 2-322-161.45-c1-0-7
Degree $2$
Conductor $322$
Sign $0.901 - 0.433i$
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 + (2.81 + 1.62i)3-s + (−0.499 + 0.866i)4-s + (1.55 + 2.68i)5-s − 3.24i·6-s + (−0.334 − 2.62i)7-s + 0.999·8-s + (3.77 + 6.54i)9-s + (1.55 − 2.68i)10-s + (−4.07 − 2.35i)11-s + (−2.81 + 1.62i)12-s − 3.22i·13-s + (−2.10 + 1.60i)14-s + 10.0i·15-s + (−0.5 − 0.866i)16-s + (0.465 − 0.806i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (1.62 + 0.937i)3-s + (−0.249 + 0.433i)4-s + (0.693 + 1.20i)5-s − 1.32i·6-s + (−0.126 − 0.991i)7-s + 0.353·8-s + (1.25 + 2.18i)9-s + (0.490 − 0.849i)10-s + (−1.22 − 0.709i)11-s + (−0.812 + 0.468i)12-s − 0.894i·13-s + (−0.562 + 0.428i)14-s + 2.60i·15-s + (−0.125 − 0.216i)16-s + (0.112 − 0.195i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.901 - 0.433i$
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: \((2,\ 322,\ (\ :1/2),\ 0.901 - 0.433i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81688 + 0.413763i\)
\(L(\frac12)\) \(\approx\) \(1.81688 + 0.413763i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.334 + 2.62i)T \)
23 \( 1 + (3.61 - 3.15i)T \)
good3 \( 1 + (-2.81 - 1.62i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.55 - 2.68i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.07 + 2.35i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.22iT - 13T^{2} \)
17 \( 1 + (-0.465 + 0.806i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.55 - 2.68i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 1.03T + 29T^{2} \)
31 \( 1 + (8.51 + 4.91i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.60 + 5.54i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.22iT - 41T^{2} \)
43 \( 1 - 0.943iT - 43T^{2} \)
47 \( 1 + (-4.29 + 2.47i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.11 - 3.52i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.109 - 0.0634i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.63 + 4.56i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0606 + 0.0349i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.66T + 71T^{2} \)
73 \( 1 + (8.51 + 4.91i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.39 - 0.806i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.34T + 83T^{2} \)
89 \( 1 + (-6.59 - 11.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.02T + 97T^{2} \)
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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

−11.03991645749202515735141569018, −10.43531824948602609883621060428, −10.03691421718132904820065940979, −9.178066986806918904372386494475, −7.81870532083548114079857570115, −7.57695437627702971409091364780, −5.60477451959020378595326413605, −3.93250591995093664852016123933, −3.15712055998557801224332102619, −2.32286186525318504779904020154, 1.64566438457182506840519986928, 2.61012244121069355594983670581, 4.59085893394312344079500713359, 5.81494327895404164841536501242, 7.00395396699973116234241858381, 7.965505683358707668488807525410, 8.731133824873230833871682837312, 9.228071640979349975450120063737, 9.985995551673907575940974442279, 12.01034345041854428402592687121

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