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Properties

Label 2-320-16.5-c1-0-2
Degree $2$
Conductor $320$
Sign $0.412 - 0.910i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.209 + 0.209i)3-s + (0.707 + 0.707i)5-s + 1.73i·7-s + 2.91i·9-s + (−0.505 − 0.505i)11-s + (−1.88 + 1.88i)13-s − 0.296·15-s + 4.53·17-s + (3.22 − 3.22i)19-s + (−0.364 − 0.364i)21-s + 8.85i·23-s + 1.00i·25-s + (−1.23 − 1.23i)27-s + (−2.44 + 2.44i)29-s + 5.70·31-s + ⋯
L(s)  = 1  + (−0.120 + 0.120i)3-s + (0.316 + 0.316i)5-s + 0.656i·7-s + 0.970i·9-s + (−0.152 − 0.152i)11-s + (−0.523 + 0.523i)13-s − 0.0765·15-s + 1.09·17-s + (0.738 − 0.738i)19-s + (−0.0794 − 0.0794i)21-s + 1.84i·23-s + 0.200i·25-s + (−0.238 − 0.238i)27-s + (−0.453 + 0.453i)29-s + 1.02·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.412 - 0.910i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.412 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06868 + 0.688999i\)
\(L(\frac12)\) \(\approx\) \(1.06868 + 0.688999i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.209 - 0.209i)T - 3iT^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
11 \( 1 + (0.505 + 0.505i)T + 11iT^{2} \)
13 \( 1 + (1.88 - 1.88i)T - 13iT^{2} \)
17 \( 1 - 4.53T + 17T^{2} \)
19 \( 1 + (-3.22 + 3.22i)T - 19iT^{2} \)
23 \( 1 - 8.85iT - 23T^{2} \)
29 \( 1 + (2.44 - 2.44i)T - 29iT^{2} \)
31 \( 1 - 5.70T + 31T^{2} \)
37 \( 1 + (5.35 + 5.35i)T + 37iT^{2} \)
41 \( 1 + 10.0iT - 41T^{2} \)
43 \( 1 + (-2.10 - 2.10i)T + 43iT^{2} \)
47 \( 1 + 4.32T + 47T^{2} \)
53 \( 1 + (1.37 + 1.37i)T + 53iT^{2} \)
59 \( 1 + (6.64 + 6.64i)T + 59iT^{2} \)
61 \( 1 + (-5.26 + 5.26i)T - 61iT^{2} \)
67 \( 1 + (-10.5 + 10.5i)T - 67iT^{2} \)
71 \( 1 + 14.0iT - 71T^{2} \)
73 \( 1 - 6.63iT - 73T^{2} \)
79 \( 1 + 4.27T + 79T^{2} \)
83 \( 1 + (9.15 - 9.15i)T - 83iT^{2} \)
89 \( 1 + 3.23iT - 89T^{2} \)
97 \( 1 - 1.94T + 97T^{2} \)
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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

−11.70748818975702809495806772690, −10.93844223983962242678741631360, −9.886543933223624145721768463492, −9.178603101336660832486356757122, −7.898723929762047837999221753671, −7.09341955887614336418471032068, −5.65242740048345255512956557603, −5.06205012130238771653088681563, −3.34437782749135898718594584948, −2.01514274984091931746978996846, 1.00887726684195848557319354552, 2.99897787494994720074301106114, 4.33074978720296491226923745281, 5.55689975986551586686658818744, 6.57437834415058821778369953502, 7.63495360604626601437902454130, 8.591844286152483937595535333593, 9.943705399251669422437913010455, 10.16825246176558374275485204691, 11.66823816494893086122104871023

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