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Properties

Label 2-322-161.10-c1-0-8
Degree $2$
Conductor $322$
Sign $0.395 + 0.918i$
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 + (0.0599 + 0.116i)3-s + (−0.786 − 0.618i)4-s + (−0.756 + 0.0722i)5-s + (0.129 − 0.0186i)6-s + (2.12 + 1.58i)7-s + (−0.841 + 0.540i)8-s + (1.73 − 2.42i)9-s + (−0.179 + 0.738i)10-s + (4.23 − 1.46i)11-s + (0.0247 − 0.128i)12-s + (−1.10 − 3.75i)13-s + (2.18 − 1.48i)14-s + (−0.0538 − 0.0837i)15-s + (0.235 + 0.971i)16-s + (−1.51 − 0.607i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (0.0346 + 0.0671i)3-s + (−0.393 − 0.309i)4-s + (−0.338 + 0.0323i)5-s + (0.0529 − 0.00760i)6-s + (0.801 + 0.598i)7-s + (−0.297 + 0.191i)8-s + (0.576 − 0.809i)9-s + (−0.0566 + 0.233i)10-s + (1.27 − 0.441i)11-s + (0.00715 − 0.0371i)12-s + (−0.305 − 1.04i)13-s + (0.585 − 0.397i)14-s + (−0.0138 − 0.0216i)15-s + (0.0589 + 0.242i)16-s + (−0.368 − 0.147i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.395 + 0.918i$
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: \((2,\ 322,\ (\ :1/2),\ 0.395 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28485 - 0.845633i\)
\(L(\frac12)\) \(\approx\) \(1.28485 - 0.845633i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 - 1.58i)T \)
23 \( 1 + (-3.65 + 3.11i)T \)
good3 \( 1 + (-0.0599 - 0.116i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (0.756 - 0.0722i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-4.23 + 1.46i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (1.10 + 3.75i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (1.51 + 0.607i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-1.22 + 0.489i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-1.26 - 8.76i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (9.12 - 0.434i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-4.53 - 3.22i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-11.2 - 5.11i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-0.695 + 1.08i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (2.57 - 1.48i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.45 + 3.62i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (4.51 + 1.09i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (9.75 + 5.03i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (0.242 + 1.25i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-0.873 - 1.00i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (0.0928 - 0.118i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (11.1 - 11.6i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (1.82 + 3.99i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.812 - 17.0i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (6.92 - 15.1i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.41517037218357125057129587449, −10.84061543257912393257318211075, −9.475953578552504020199266201754, −8.933456361554442496146031252009, −7.73957008411968785054623378918, −6.47632417800984495339232060518, −5.26972004498413743928298467010, −4.14809833487246322714345220861, −3.04426525434511894559108894664, −1.28320193516640905484905223841, 1.78462965174681167989592046793, 4.10415209835293184814022501325, 4.49744899323043259393797550611, 5.97386596210113645075281144386, 7.35279838248621291943568262977, 7.51223625217781992324300023489, 8.911703606389818975246223036098, 9.739886996077680333648616253562, 11.09074970878507412712284763220, 11.72724926136219051768061635568

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