Properties

Label 2-320-5.4-c3-0-11
Degree $2$
Conductor $320$
Sign $-0.447 - 0.894i$
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  + 2i·3-s + (5 + 10i)5-s + 26i·7-s + 23·9-s + 28·11-s + 12i·13-s + (−20 + 10i)15-s − 64i·17-s − 60·19-s − 52·21-s + 58i·23-s + (−75 + 100i)25-s + 100i·27-s + 90·29-s − 128·31-s + ⋯
L(s)  = 1  + 0.384i·3-s + (0.447 + 0.894i)5-s + 1.40i·7-s + 0.851·9-s + 0.767·11-s + 0.256i·13-s + (−0.344 + 0.172i)15-s − 0.913i·17-s − 0.724·19-s − 0.540·21-s + 0.525i·23-s + (−0.599 + 0.800i)25-s + 0.712i·27-s + 0.576·29-s − 0.741·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.447 - 0.894i$
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: \((2,\ 320,\ (\ :3/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.037591275\)
\(L(\frac12)\) \(\approx\) \(2.037591275\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5 - 10i)T \)
good3 \( 1 - 2iT - 27T^{2} \)
7 \( 1 - 26iT - 343T^{2} \)
11 \( 1 - 28T + 1.33e3T^{2} \)
13 \( 1 - 12iT - 2.19e3T^{2} \)
17 \( 1 + 64iT - 4.91e3T^{2} \)
19 \( 1 + 60T + 6.85e3T^{2} \)
23 \( 1 - 58iT - 1.21e4T^{2} \)
29 \( 1 - 90T + 2.43e4T^{2} \)
31 \( 1 + 128T + 2.97e4T^{2} \)
37 \( 1 + 236iT - 5.06e4T^{2} \)
41 \( 1 - 242T + 6.89e4T^{2} \)
43 \( 1 - 362iT - 7.95e4T^{2} \)
47 \( 1 - 226iT - 1.03e5T^{2} \)
53 \( 1 + 108iT - 1.48e5T^{2} \)
59 \( 1 + 20T + 2.05e5T^{2} \)
61 \( 1 + 542T + 2.26e5T^{2} \)
67 \( 1 - 434iT - 3.00e5T^{2} \)
71 \( 1 + 1.12e3T + 3.57e5T^{2} \)
73 \( 1 + 632iT - 3.89e5T^{2} \)
79 \( 1 - 720T + 4.93e5T^{2} \)
83 \( 1 + 478iT - 5.71e5T^{2} \)
89 \( 1 - 490T + 7.04e5T^{2} \)
97 \( 1 - 1.45e3iT - 9.12e5T^{2} \)
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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

−11.44882135083684824148929030341, −10.57414412333405307026209586741, −9.436179251195188422269726643836, −9.119870743668240181451128152218, −7.56716214727069016926041410952, −6.55304683624049432613562631405, −5.68297296987992963923648045740, −4.39864792936923166987060598602, −3.02336676388724617027351559708, −1.84391091884897757701568395479, 0.77660257682822605928917914978, 1.75232921695408700796812412207, 3.89482963891029657179100245135, 4.61104182251339335596954819002, 6.13574437147212438980891751340, 7.00129800582575677014624815039, 8.010867569235655360101214641531, 8.995261770823402169029000634345, 10.11556071257465796326139075582, 10.65293564668183674781522240351

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