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Properties

Label 2-322-161.10-c1-0-14
Degree $2$
Conductor $322$
Sign $-0.826 - 0.563i$
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  + (0.327 − 0.945i)2-s + (−0.132 − 0.257i)3-s + (−0.786 − 0.618i)4-s + (−2.50 + 0.239i)5-s + (−0.286 + 0.0412i)6-s + (−2.51 + 0.810i)7-s + (−0.841 + 0.540i)8-s + (1.69 − 2.37i)9-s + (−0.594 + 2.45i)10-s + (−4.41 + 1.52i)11-s + (−0.0548 + 0.284i)12-s + (0.296 + 1.01i)13-s + (−0.0579 + 2.64i)14-s + (0.394 + 0.614i)15-s + (0.235 + 0.971i)16-s + (−3.26 − 1.30i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (−0.0766 − 0.148i)3-s + (−0.393 − 0.309i)4-s + (−1.12 + 0.107i)5-s + (−0.117 + 0.0168i)6-s + (−0.951 + 0.306i)7-s + (−0.297 + 0.191i)8-s + (0.563 − 0.791i)9-s + (−0.187 + 0.774i)10-s + (−1.33 + 0.460i)11-s + (−0.0158 + 0.0821i)12-s + (0.0822 + 0.280i)13-s + (−0.0154 + 0.706i)14-s + (0.101 + 0.158i)15-s + (0.0589 + 0.242i)16-s + (−0.791 − 0.316i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.826 - 0.563i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.826 - 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0540709 + 0.175374i\)
\(L(\frac12)\) \(\approx\) \(0.0540709 + 0.175374i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.327 + 0.945i)T \)
7 \( 1 + (2.51 - 0.810i)T \)
23 \( 1 + (-4.55 + 1.50i)T \)
good3 \( 1 + (0.132 + 0.257i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (2.50 - 0.239i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (4.41 - 1.52i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.296 - 1.01i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (3.26 + 1.30i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (4.41 - 1.76i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (1.00 + 6.95i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-2.13 + 0.101i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-0.561 - 0.399i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-0.811 - 0.370i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-1.84 + 2.87i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (2.33 - 1.34i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.44 + 7.80i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (10.6 + 2.58i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-5.53 - 2.85i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-0.222 - 1.15i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-5.61 - 6.47i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (7.55 - 9.60i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-4.48 + 4.70i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-0.484 - 1.06i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.700 + 14.7i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (0.741 - 1.62i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.21233856103450859360180364241, −10.23727440589088371400012798529, −9.392451596493173667099912914588, −8.287094555811094972914427866340, −7.16119362218793778962661788243, −6.19610879354660043636471600814, −4.65966704734165908758457935618, −3.72366255788615969693206537223, −2.50150318815757628932538534370, −0.11660099258768322379183191156, 3.00559788701874223305891186342, 4.21449051978265450751465790752, 5.13202249252846456981692672018, 6.47715820070005461205017127622, 7.45841487152809539924937043283, 8.134103356396686225051608767348, 9.182746888249871530173162850633, 10.57285335882827198570851507651, 10.99050408386279762147816392697, 12.54667558770901055911229260866

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