Properties

Label 2-322-161.45-c1-0-9
Degree $2$
Conductor $322$
Sign $-0.195 + 0.980i$
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 + (−0.508 − 0.293i)3-s + (−0.499 + 0.866i)4-s + (0.863 + 1.49i)5-s + 0.587i·6-s + (−1.63 − 2.08i)7-s + 0.999·8-s + (−1.32 − 2.29i)9-s + (0.863 − 1.49i)10-s + (3.46 + 2.00i)11-s + (0.508 − 0.293i)12-s − 5.60i·13-s + (−0.987 + 2.45i)14-s − 1.01i·15-s + (−0.5 − 0.866i)16-s + (2.91 − 5.05i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.293 − 0.169i)3-s + (−0.249 + 0.433i)4-s + (0.386 + 0.668i)5-s + 0.239i·6-s + (−0.616 − 0.787i)7-s + 0.353·8-s + (−0.442 − 0.766i)9-s + (0.273 − 0.472i)10-s + (1.04 + 0.603i)11-s + (0.146 − 0.0848i)12-s − 1.55i·13-s + (−0.263 + 0.656i)14-s − 0.262i·15-s + (−0.125 − 0.216i)16-s + (0.708 − 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.195 + 0.980i$
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: \((2,\ 322,\ (\ :1/2),\ -0.195 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.581471 - 0.709095i\)
\(L(\frac12)\) \(\approx\) \(0.581471 - 0.709095i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 + 2.08i)T \)
23 \( 1 + (-2.15 + 4.28i)T \)
good3 \( 1 + (0.508 + 0.293i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.863 - 1.49i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.46 - 2.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.60iT - 13T^{2} \)
17 \( 1 + (-2.91 + 5.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.863 - 1.49i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 9.71T + 29T^{2} \)
31 \( 1 + (-3.47 - 2.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.43 - 4.29i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.60iT - 41T^{2} \)
43 \( 1 - 3.11iT - 43T^{2} \)
47 \( 1 + (-6.35 + 3.66i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.09 - 1.20i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.45 - 3.14i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.19 - 2.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.25 - 5.34i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + (-3.47 - 2.00i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.75 - 5.05i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + (7.80 + 13.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−11.31576332898212879894391151309, −10.28563576268643415464445798008, −9.828539227342183361482929380353, −8.774537573309437155439204756813, −7.34743476737733136820603046394, −6.69618520400456636093793681068, −5.46762955438429436628146421376, −3.76069308529121319011823582678, −2.86046675969596192243155773858, −0.816065964151386762035711075088, 1.75106627586566669586479221598, 3.82103371482868153081493568975, 5.29566868964345422971860218254, 5.91205889060242267469186279238, 6.93783270515988480350331573037, 8.325264141785479935326996653557, 9.138274596109313523370146007471, 9.589163654547954633693070729770, 11.01848668064269117190364467453, 11.75232589799705725419348319113

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