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Properties

Label 2-320-16.13-c1-0-4
Degree $2$
Conductor $320$
Sign $0.823 + 0.567i$
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  + (0.120 + 0.120i)3-s + (−0.707 + 0.707i)5-s − 2.66i·7-s − 2.97i·9-s + (3.49 − 3.49i)11-s + (2.94 + 2.94i)13-s − 0.169·15-s + 1.85·17-s + (3.44 + 3.44i)19-s + (0.320 − 0.320i)21-s + 0.707i·23-s − 1.00i·25-s + (0.716 − 0.716i)27-s + (−3.49 − 3.49i)29-s − 6.84·31-s + ⋯
L(s)  = 1  + (0.0692 + 0.0692i)3-s + (−0.316 + 0.316i)5-s − 1.00i·7-s − 0.990i·9-s + (1.05 − 1.05i)11-s + (0.815 + 0.815i)13-s − 0.0438·15-s + 0.448·17-s + (0.791 + 0.791i)19-s + (0.0698 − 0.0698i)21-s + 0.147i·23-s − 0.200i·25-s + (0.137 − 0.137i)27-s + (−0.649 − 0.649i)29-s − 1.22·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.823 + 0.567i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.823 + 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28844 - 0.401115i\)
\(L(\frac12)\) \(\approx\) \(1.28844 - 0.401115i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.120 - 0.120i)T + 3iT^{2} \)
7 \( 1 + 2.66iT - 7T^{2} \)
11 \( 1 + (-3.49 + 3.49i)T - 11iT^{2} \)
13 \( 1 + (-2.94 - 2.94i)T + 13iT^{2} \)
17 \( 1 - 1.85T + 17T^{2} \)
19 \( 1 + (-3.44 - 3.44i)T + 19iT^{2} \)
23 \( 1 - 0.707iT - 23T^{2} \)
29 \( 1 + (3.49 + 3.49i)T + 29iT^{2} \)
31 \( 1 + 6.84T + 31T^{2} \)
37 \( 1 + (0.0975 - 0.0975i)T - 37iT^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (4.43 - 4.43i)T - 43iT^{2} \)
47 \( 1 - 1.89T + 47T^{2} \)
53 \( 1 + (7.43 - 7.43i)T - 53iT^{2} \)
59 \( 1 + (0.959 - 0.959i)T - 59iT^{2} \)
61 \( 1 + (-6.49 - 6.49i)T + 61iT^{2} \)
67 \( 1 + (3.49 + 3.49i)T + 67iT^{2} \)
71 \( 1 + 7.86iT - 71T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 - 6.70T + 79T^{2} \)
83 \( 1 + (-3.87 - 3.87i)T + 83iT^{2} \)
89 \( 1 - 10.5iT - 89T^{2} \)
97 \( 1 - 4.79T + 97T^{2} \)
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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

−11.50166468554466814164290578059, −10.76479108759242892392843996523, −9.592444218706101804348198402481, −8.846092626884892945997732824501, −7.63994376308430008036770977749, −6.69044738608127509159295024691, −5.82132784169724327800946405158, −3.86197326305046607914012680490, −3.62293611097328194735589141309, −1.15074454850874304499644749651, 1.75729041246059039467255076696, 3.31944586133205545752921006021, 4.79224789040547057847700486449, 5.65199840075229104124058626327, 7.00256682925892739464308888166, 7.986300561420916214506214325540, 8.902209159248650278254577529098, 9.715983366568883254721665317449, 10.95396411564479086229928326098, 11.73896697098639403859454322134

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