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Properties

Label 2-320-80.13-c2-0-13
Degree $2$
Conductor $320$
Sign $0.315 + 0.949i$
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  + 0.119i·3-s + (−3.85 + 3.18i)5-s + (−4.73 − 4.73i)7-s + 8.98·9-s + (−2.49 + 2.49i)11-s − 18.4i·13-s + (−0.380 − 0.462i)15-s + (10.7 − 10.7i)17-s + (−0.469 + 0.469i)19-s + (0.567 − 0.567i)21-s + (27.9 − 27.9i)23-s + (4.76 − 24.5i)25-s + 2.15i·27-s + (15.6 − 15.6i)29-s − 49.2·31-s + ⋯
L(s)  = 1  + 0.0399i·3-s + (−0.771 + 0.636i)5-s + (−0.677 − 0.677i)7-s + 0.998·9-s + (−0.226 + 0.226i)11-s − 1.41i·13-s + (−0.0253 − 0.0308i)15-s + (0.630 − 0.630i)17-s + (−0.0247 + 0.0247i)19-s + (0.0270 − 0.0270i)21-s + (1.21 − 1.21i)23-s + (0.190 − 0.981i)25-s + 0.0797i·27-s + (0.538 − 0.538i)29-s − 1.58·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 + 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.315 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.315 + 0.949i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ 0.315 + 0.949i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.935874 - 0.675386i\)
\(L(\frac12)\) \(\approx\) \(0.935874 - 0.675386i\)
\(L(2)\) 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 \( 1 + (3.85 - 3.18i)T \)
good3 \( 1 - 0.119iT - 9T^{2} \)
7 \( 1 + (4.73 + 4.73i)T + 49iT^{2} \)
11 \( 1 + (2.49 - 2.49i)T - 121iT^{2} \)
13 \( 1 + 18.4iT - 169T^{2} \)
17 \( 1 + (-10.7 + 10.7i)T - 289iT^{2} \)
19 \( 1 + (0.469 - 0.469i)T - 361iT^{2} \)
23 \( 1 + (-27.9 + 27.9i)T - 529iT^{2} \)
29 \( 1 + (-15.6 + 15.6i)T - 841iT^{2} \)
31 \( 1 + 49.2T + 961T^{2} \)
37 \( 1 - 29.1iT - 1.36e3T^{2} \)
41 \( 1 + 16.4iT - 1.68e3T^{2} \)
43 \( 1 - 4.48T + 1.84e3T^{2} \)
47 \( 1 + (-40.6 + 40.6i)T - 2.20e3iT^{2} \)
53 \( 1 + 78.7T + 2.80e3T^{2} \)
59 \( 1 + (3.26 + 3.26i)T + 3.48e3iT^{2} \)
61 \( 1 + (23.0 + 23.0i)T + 3.72e3iT^{2} \)
67 \( 1 - 58.9T + 4.48e3T^{2} \)
71 \( 1 + 76.8iT - 5.04e3T^{2} \)
73 \( 1 + (-18.7 + 18.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 141. iT - 6.24e3T^{2} \)
83 \( 1 - 120. iT - 6.88e3T^{2} \)
89 \( 1 + 87.7T + 7.92e3T^{2} \)
97 \( 1 + (-34.6 + 34.6i)T - 9.40e3iT^{2} \)
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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

−10.95723516255426534806403225496, −10.41067028210609131276151346260, −9.616460903809998252313695602406, −8.143334177381177747022487176825, −7.31158602624933675135934889158, −6.65554799872061452003135544534, −5.09526666863185229715820864662, −3.86068495196711422682806950331, −2.90048920241319667197389925209, −0.59339863132545363808266729150, 1.49359312753594518217876739405, 3.37496116298770129867147364514, 4.43268338067071091278256576082, 5.61256769742983395544614124344, 6.89501800607668762013005664531, 7.71529390101050304274983221804, 9.029720969344425088240119511901, 9.408748468082398257900296318545, 10.78962441367924779231516078190, 11.70054444908704814677374658499

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