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Properties

Label 2-320-5.4-c3-0-18
Degree $2$
Conductor $320$
Sign $0.983 - 0.178i$
Analytic cond. $18.8806$
Root an. cond. $4.34518$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11 − 2i)5-s + 27·9-s + 92i·13-s − 104i·17-s + (117 − 44i)25-s + 130·29-s + 396i·37-s + 230·41-s + (297 − 54i)45-s + 343·49-s − 572i·53-s + 830·61-s + (184 + 1.01e3i)65-s + 592i·73-s + 729·81-s + ⋯
L(s)  = 1  + (0.983 − 0.178i)5-s + 9-s + 1.96i·13-s − 1.48i·17-s + (0.936 − 0.351i)25-s + 0.832·29-s + 1.75i·37-s + 0.876·41-s + (0.983 − 0.178i)45-s + 49-s − 1.48i·53-s + 1.74·61-s + (0.351 + 1.93i)65-s + 0.949i·73-s + 0.999·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.983 - 0.178i$
Analytic conductor: \(18.8806\)
Root analytic conductor: \(4.34518\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :3/2),\ 0.983 - 0.178i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.426969299\)
\(L(\frac12)\) \(\approx\) \(2.426969299\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11 + 2i)T \)
good3 \( 1 - 27T^{2} \)
7 \( 1 - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 92iT - 2.19e3T^{2} \)
17 \( 1 + 104iT - 4.91e3T^{2} \)
19 \( 1 + 6.85e3T^{2} \)
23 \( 1 - 1.21e4T^{2} \)
29 \( 1 - 130T + 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 - 396iT - 5.06e4T^{2} \)
41 \( 1 - 230T + 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 - 1.03e5T^{2} \)
53 \( 1 + 572iT - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 830T + 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 592iT - 3.89e5T^{2} \)
79 \( 1 + 4.93e5T^{2} \)
83 \( 1 - 5.71e5T^{2} \)
89 \( 1 + 1.67e3T + 7.04e5T^{2} \)
97 \( 1 + 1.81e3iT - 9.12e5T^{2} \)
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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

−11.30998510555258313307458755969, −9.987836193224612094693890950500, −9.553716625062177340444559116481, −8.605416351110581546438675354429, −7.07496566472422275325860357065, −6.55793097876415324198428815531, −5.12210525051756329349756107597, −4.24969900961530370403976375404, −2.47041010066281396351503514633, −1.27113362172153832304053666927, 1.10795957326549134486653797820, 2.52544736311374169812265038982, 3.93774191750573984236258134170, 5.37616669333440697868220933832, 6.14292864079838496245629492831, 7.32954636025287382537985767739, 8.314249022432188667688014427833, 9.467293736291278086317177880099, 10.46530810280346546523870865482, 10.66245508300188475930443054229

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