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Properties

Label 2-320-40.29-c1-0-11
Degree $2$
Conductor $320$
Sign $-0.707 + 0.707i$
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  − 2.23·5-s − 4.47i·7-s − 3·9-s + 2i·11-s − 4.47·13-s − 6i·19-s − 4.47i·23-s + 5.00·25-s + 10.0i·35-s − 4.47·37-s + 2·41-s + 6.70·45-s + 13.4i·47-s − 13.0·49-s + 13.4·53-s + ⋯
L(s)  = 1  − 0.999·5-s − 1.69i·7-s − 9-s + 0.603i·11-s − 1.24·13-s − 1.37i·19-s − 0.932i·23-s + 1.00·25-s + 1.69i·35-s − 0.735·37-s + 0.312·41-s + 0.999·45-s + 1.95i·47-s − 1.85·49-s + 1.84·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.214616 - 0.518130i\)
\(L(\frac12)\) \(\approx\) \(0.214616 - 0.518130i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T \)
good3 \( 1 + 3T^{2} \)
7 \( 1 + 4.47iT - 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 4.47iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 13.4iT - 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + 14iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.19095231951279286020364948773, −10.56136632464407735050871933584, −9.464143784383224213805660866195, −8.273235517327712159386382596749, −7.36117283476782268155519649770, −6.78773385258931231824529573579, −4.93953054081686779578491434614, −4.17784388377032040855948361174, −2.83092772990423054658736886172, −0.38329903122714331145618322510, 2.45746584430386316812710546766, 3.56692841494322765161433522685, 5.22023496169602041334162095360, 5.86913560571097498187262524198, 7.34280393125255434375117540885, 8.395522189463721427432451945155, 8.879967857585108203739735639697, 10.11312107605266051592590794003, 11.42176006962823845626021208649, 11.94899967580580442419799520115

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