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Properties

Label 2-320-80.53-c2-0-16
Degree $2$
Conductor $320$
Sign $0.993 + 0.109i$
Analytic cond. $8.71936$
Root an. cond. $2.95285$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.94·3-s + (3.69 − 3.37i)5-s + (3.22 + 3.22i)7-s + 15.4·9-s + (7.67 + 7.67i)11-s − 22.2·13-s + (18.2 − 16.6i)15-s + (−3.76 + 3.76i)17-s + (−0.809 − 0.809i)19-s + (15.9 + 15.9i)21-s + (12.2 − 12.2i)23-s + (2.23 − 24.8i)25-s + 31.8·27-s + (−27.1 − 27.1i)29-s − 25.5·31-s + ⋯
L(s)  = 1  + 1.64·3-s + (0.738 − 0.674i)5-s + (0.460 + 0.460i)7-s + 1.71·9-s + (0.697 + 0.697i)11-s − 1.70·13-s + (1.21 − 1.11i)15-s + (−0.221 + 0.221i)17-s + (−0.0426 − 0.0426i)19-s + (0.759 + 0.759i)21-s + (0.534 − 0.534i)23-s + (0.0895 − 0.995i)25-s + 1.18·27-s + (−0.936 − 0.936i)29-s − 0.824·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.109i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.993 + 0.109i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ 0.993 + 0.109i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.17811 - 0.174802i\)
\(L(\frac12)\) \(\approx\) \(3.17811 - 0.174802i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-3.69 + 3.37i)T \)
good3 \( 1 - 4.94T + 9T^{2} \)
7 \( 1 + (-3.22 - 3.22i)T + 49iT^{2} \)
11 \( 1 + (-7.67 - 7.67i)T + 121iT^{2} \)
13 \( 1 + 22.2T + 169T^{2} \)
17 \( 1 + (3.76 - 3.76i)T - 289iT^{2} \)
19 \( 1 + (0.809 + 0.809i)T + 361iT^{2} \)
23 \( 1 + (-12.2 + 12.2i)T - 529iT^{2} \)
29 \( 1 + (27.1 + 27.1i)T + 841iT^{2} \)
31 \( 1 + 25.5T + 961T^{2} \)
37 \( 1 - 8.62T + 1.36e3T^{2} \)
41 \( 1 - 73.4iT - 1.68e3T^{2} \)
43 \( 1 - 17.5iT - 1.84e3T^{2} \)
47 \( 1 + (17.4 - 17.4i)T - 2.20e3iT^{2} \)
53 \( 1 + 29.3iT - 2.80e3T^{2} \)
59 \( 1 + (-72.6 + 72.6i)T - 3.48e3iT^{2} \)
61 \( 1 + (-19.7 + 19.7i)T - 3.72e3iT^{2} \)
67 \( 1 - 14.0iT - 4.48e3T^{2} \)
71 \( 1 - 91.3iT - 5.04e3T^{2} \)
73 \( 1 + (-42.7 + 42.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 46.7iT - 6.24e3T^{2} \)
83 \( 1 + 82.9T + 6.88e3T^{2} \)
89 \( 1 + 131.T + 7.92e3T^{2} \)
97 \( 1 + (30.5 - 30.5i)T - 9.40e3iT^{2} \)
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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

−11.53636262085734712585709779723, −9.754735859207725121268833929556, −9.627804783586297538767710148266, −8.638044387503135598975284818947, −7.84668071864939824490904876898, −6.77783487780700093389252373452, −5.15948156590659606036076991387, −4.20693009866024374819191165309, −2.57238485327745143474660181708, −1.80774148425681554941210478705, 1.80947731468932872640524190661, 2.85912908352064914511099970611, 3.91049523640927053171611918570, 5.39427778788023103480561210757, 7.07830618854302907457539146096, 7.45489725913716307928371457607, 8.815862761814947270510796891107, 9.382660582127121918028410088827, 10.27324033965308739502880972417, 11.26545997814855201461981749734

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