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Properties

Label 2-320-40.29-c5-0-18
Degree $2$
Conductor $320$
Sign $0.347 - 0.937i$
Analytic cond. $51.3228$
Root an. cond. $7.16399$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11.4·3-s + (−32.3 + 45.6i)5-s + 121. i·7-s − 113.·9-s − 625. i·11-s + 924.·13-s + (368. − 520. i)15-s − 1.32e3i·17-s + 1.13e3i·19-s − 1.38e3i·21-s − 3.05e3i·23-s + (−1.03e3 − 2.94e3i)25-s + 4.05e3·27-s + 8.23e3i·29-s + 1.98e3·31-s + ⋯
L(s)  = 1  − 0.731·3-s + (−0.578 + 0.816i)5-s + 0.936i·7-s − 0.465·9-s − 1.55i·11-s + 1.51·13-s + (0.422 − 0.596i)15-s − 1.11i·17-s + 0.721i·19-s − 0.684i·21-s − 1.20i·23-s + (−0.331 − 0.943i)25-s + 1.07·27-s + 1.81i·29-s + 0.371·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.347 - 0.937i$
Analytic conductor: \(51.3228\)
Root analytic conductor: \(7.16399\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ 0.347 - 0.937i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.078375237\)
\(L(\frac12)\) \(\approx\) \(1.078375237\)
\(L(\frac{7}{2})\) 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 \( 1 + (32.3 - 45.6i)T \)
good3 \( 1 + 11.4T + 243T^{2} \)
7 \( 1 - 121. iT - 1.68e4T^{2} \)
11 \( 1 + 625. iT - 1.61e5T^{2} \)
13 \( 1 - 924.T + 3.71e5T^{2} \)
17 \( 1 + 1.32e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.13e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.05e3iT - 6.43e6T^{2} \)
29 \( 1 - 8.23e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.98e3T + 2.86e7T^{2} \)
37 \( 1 + 9.68e3T + 6.93e7T^{2} \)
41 \( 1 + 1.58e4T + 1.15e8T^{2} \)
43 \( 1 - 1.87e4T + 1.47e8T^{2} \)
47 \( 1 + 5.16e3iT - 2.29e8T^{2} \)
53 \( 1 - 5.65e3T + 4.18e8T^{2} \)
59 \( 1 - 2.86e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.41e3iT - 8.44e8T^{2} \)
67 \( 1 - 4.58e4T + 1.35e9T^{2} \)
71 \( 1 - 5.53e3T + 1.80e9T^{2} \)
73 \( 1 - 5.88e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.93e4T + 3.07e9T^{2} \)
83 \( 1 + 1.02e5T + 3.93e9T^{2} \)
89 \( 1 + 9.52e4T + 5.58e9T^{2} \)
97 \( 1 - 5.52e4iT - 8.58e9T^{2} \)
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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

−11.06088112613721362298718503511, −10.48845161021160637486685847487, −8.707431000753820850445919185056, −8.469402058146002843996320387089, −6.85525267500431021753326590324, −6.04665658535531749259083700549, −5.31690221730340067553300626069, −3.60894845441492569348912247715, −2.77015359721821143369826631797, −0.791553201967535539842119489294, 0.47249389737628958432641094799, 1.61311287747234486762581514700, 3.72117409411401976800319586549, 4.50451190616763123647186633181, 5.61334433572892032607885051325, 6.70640201619931862408701605543, 7.76152460713504150209890178862, 8.644919403303540410873445674565, 9.788276856460303236631014244157, 10.79621536226579908511630843746

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