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Properties

Label 2-320-40.29-c1-0-4
Degree $2$
Conductor $320$
Sign $0.584 + 0.811i$
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  − 2.44·3-s + (−1.41 + 1.73i)5-s − 1.41i·7-s + 2.99·9-s − 2i·11-s + 5.65·13-s + (3.46 − 4.24i)15-s − 4.89i·17-s − 6i·19-s + 3.46i·21-s + 7.07i·23-s + (−0.999 − 4.89i)25-s − 6.92i·29-s + 6.92·31-s + 4.89i·33-s + ⋯
L(s)  = 1  − 1.41·3-s + (−0.632 + 0.774i)5-s − 0.534i·7-s + 0.999·9-s − 0.603i·11-s + 1.56·13-s + (0.894 − 1.09i)15-s − 1.18i·17-s − 1.37i·19-s + 0.755i·21-s + 1.47i·23-s + (−0.199 − 0.979i)25-s − 1.28i·29-s + 1.24·31-s + 0.852i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.584 + 0.811i$
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: \((2,\ 320,\ (\ :1/2),\ 0.584 + 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.588737 - 0.301476i\)
\(L(\frac12)\) \(\approx\) \(0.588737 - 0.301476i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - 1.73i)T \)
good3 \( 1 + 2.44T + 3T^{2} \)
7 \( 1 + 1.41iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 - 7.07iT - 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 + 2.82T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 2.44T + 43T^{2} \)
47 \( 1 + 4.24iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 2iT - 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 - 2.44T + 67T^{2} \)
71 \( 1 + 6.92T + 71T^{2} \)
73 \( 1 + 4.89iT - 73T^{2} \)
79 \( 1 + 6.92T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 14.6iT - 97T^{2} \)
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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

−11.45160167584911197455789172559, −10.91752135931185007087567807877, −9.994985523808342936963889936037, −8.598163691516383065625469503059, −7.34995260832987264352652700661, −6.57771911182382030034896459962, −5.68777646897318915323387455516, −4.44542507118119795734508654157, −3.21517462654926945323503370272, −0.65906081523396271103566142236, 1.33870051875073351926471888572, 3.81272608323679950141687032036, 4.86721350314232118735051269196, 5.87365554198466454328991910957, 6.59459632149574036899883765979, 8.151115769539026665539807118465, 8.762545750106852519057842399286, 10.28223084200974468655470455457, 10.89146641442965994820938700792, 11.96945144208321185361131909053

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