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Properties

Label 2-322-161.68-c1-0-9
Degree $2$
Conductor $322$
Sign $0.523 + 0.852i$
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 + (−1.07 + 0.622i)3-s + (−0.499 − 0.866i)4-s + (0.668 − 1.15i)5-s + 1.24i·6-s + (2.09 + 1.61i)7-s − 0.999·8-s + (−0.726 + 1.25i)9-s + (−0.668 − 1.15i)10-s + (5.22 − 3.01i)11-s + (1.07 + 0.622i)12-s − 6.52i·13-s + (2.44 − 1.00i)14-s + 1.66i·15-s + (−0.5 + 0.866i)16-s + (0.568 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.622 + 0.359i)3-s + (−0.249 − 0.433i)4-s + (0.299 − 0.518i)5-s + 0.507i·6-s + (0.791 + 0.610i)7-s − 0.353·8-s + (−0.242 + 0.419i)9-s + (−0.211 − 0.366i)10-s + (1.57 − 0.909i)11-s + (0.311 + 0.179i)12-s − 1.80i·13-s + (0.653 − 0.269i)14-s + 0.429i·15-s + (−0.125 + 0.216i)16-s + (0.137 + 0.238i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.523 + 0.852i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.523 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26717 - 0.709049i\)
\(L(\frac12)\) \(\approx\) \(1.26717 - 0.709049i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.09 - 1.61i)T \)
23 \( 1 + (3.30 - 3.47i)T \)
good3 \( 1 + (1.07 - 0.622i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.668 + 1.15i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.22 + 3.01i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.52iT - 13T^{2} \)
17 \( 1 + (-0.568 - 0.984i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.77 + 3.07i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 3.18T + 29T^{2} \)
31 \( 1 + (-2.80 + 1.61i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.10 - 2.94i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.928iT - 41T^{2} \)
43 \( 1 + 0.303iT - 43T^{2} \)
47 \( 1 + (0.243 + 0.140i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.3 - 6.56i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.10 - 0.635i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.00 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.57 + 2.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.02T + 71T^{2} \)
73 \( 1 + (11.1 - 6.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.6 - 6.13i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.99T + 83T^{2} \)
89 \( 1 + (-4.71 + 8.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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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

−11.42407881596843990801975325288, −10.88424114421647960814018085139, −9.737179217548897047008148879820, −8.818582490626723798886803948507, −7.891524525605741763663812017573, −5.94093961678621338929944139456, −5.52358067598427184393598355199, −4.47130801728886810442530782721, −3.02407497183484540445419981071, −1.24105577694972906786792514004, 1.69259765303818018665724547660, 3.87659307439006000528528857217, 4.72722028962022985504754364122, 6.32066623713765016562472966328, 6.62783724253815669425718596861, 7.59828300849366624522430272415, 8.966115179302489729538658393274, 9.803993326709640203928168712483, 11.18674303669703425432404359815, 11.84524243137044670355028123449

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