Properties

Label 2-320-5.2-c4-0-26
Degree $2$
Conductor $320$
Sign $0.875 + 0.483i$
Analytic cond. $33.0783$
Root an. cond. $5.75138$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.26 − 5.26i)3-s + (−21.7 + 12.2i)5-s + (−4.21 − 4.21i)7-s + 25.6i·9-s − 152.·11-s + (136. − 136. i)13-s + (−50.1 + 179. i)15-s + (348. + 348. i)17-s − 527. i·19-s − 44.3·21-s + (70.5 − 70.5i)23-s + (324. − 534. i)25-s + (561. + 561. i)27-s − 68.9i·29-s + 1.37e3·31-s + ⋯
L(s)  = 1  + (0.584 − 0.584i)3-s + (−0.871 + 0.490i)5-s + (−0.0859 − 0.0859i)7-s + 0.316i·9-s − 1.26·11-s + (0.808 − 0.808i)13-s + (−0.222 + 0.796i)15-s + (1.20 + 1.20i)17-s − 1.46i·19-s − 0.100·21-s + (0.133 − 0.133i)23-s + (0.518 − 0.854i)25-s + (0.769 + 0.769i)27-s − 0.0820i·29-s + 1.42·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.875 + 0.483i$
Analytic conductor: \(33.0783\)
Root analytic conductor: \(5.75138\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :2),\ 0.875 + 0.483i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.954642126\)
\(L(\frac12)\) \(\approx\) \(1.954642126\)
\(L(3)\) 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 \( 1 + (21.7 - 12.2i)T \)
good3 \( 1 + (-5.26 + 5.26i)T - 81iT^{2} \)
7 \( 1 + (4.21 + 4.21i)T + 2.40e3iT^{2} \)
11 \( 1 + 152.T + 1.46e4T^{2} \)
13 \( 1 + (-136. + 136. i)T - 2.85e4iT^{2} \)
17 \( 1 + (-348. - 348. i)T + 8.35e4iT^{2} \)
19 \( 1 + 527. iT - 1.30e5T^{2} \)
23 \( 1 + (-70.5 + 70.5i)T - 2.79e5iT^{2} \)
29 \( 1 + 68.9iT - 7.07e5T^{2} \)
31 \( 1 - 1.37e3T + 9.23e5T^{2} \)
37 \( 1 + (-1.00e3 - 1.00e3i)T + 1.87e6iT^{2} \)
41 \( 1 - 663.T + 2.82e6T^{2} \)
43 \( 1 + (-1.63e3 + 1.63e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (438. + 438. i)T + 4.87e6iT^{2} \)
53 \( 1 + (712. - 712. i)T - 7.89e6iT^{2} \)
59 \( 1 - 2.91e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.39e3T + 1.38e7T^{2} \)
67 \( 1 + (1.69e3 + 1.69e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 3.28e3T + 2.54e7T^{2} \)
73 \( 1 + (1.46e3 - 1.46e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 3.38e3iT - 3.89e7T^{2} \)
83 \( 1 + (-8.69e3 + 8.69e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 4.27e3iT - 6.27e7T^{2} \)
97 \( 1 + (-7.21e3 - 7.21e3i)T + 8.85e7iT^{2} \)
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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

−10.73397303845016903763479718403, −10.33285821753861552363671129739, −8.630704007387237250934798671366, −7.970096145613151835362450328407, −7.40964770191396616951765954075, −6.14882897159270495007120635211, −4.82862814765910555963959793858, −3.36508641442856143882722893809, −2.54379829246586137827586003190, −0.77316942188490449945901930162, 0.912679320483465385135533011332, 2.88724993421903610588975718952, 3.82666846343109911032712522701, 4.83310691464683228407596473385, 6.07553410658913642667834877025, 7.60732553612819529493177040956, 8.193958097854061677866922354698, 9.235188768622515860030582731008, 9.950161038539127401686687716446, 11.08605406200777371615969831838

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