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Properties

Label 2-320-16.13-c3-0-19
Degree $2$
Conductor $320$
Sign $-0.685 + 0.728i$
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  + (1.80 + 1.80i)3-s + (−3.53 + 3.53i)5-s + 5.36i·7-s − 20.5i·9-s + (−26.5 + 26.5i)11-s + (−45.3 − 45.3i)13-s − 12.7·15-s − 11.5·17-s + (13.0 + 13.0i)19-s + (−9.66 + 9.66i)21-s − 33.7i·23-s − 25.0i·25-s + (85.6 − 85.6i)27-s + (−187. − 187. i)29-s − 18.8·31-s + ⋯
L(s)  = 1  + (0.346 + 0.346i)3-s + (−0.316 + 0.316i)5-s + 0.289i·7-s − 0.759i·9-s + (−0.726 + 0.726i)11-s + (−0.966 − 0.966i)13-s − 0.219·15-s − 0.165·17-s + (0.157 + 0.157i)19-s + (−0.100 + 0.100i)21-s − 0.305i·23-s − 0.200i·25-s + (0.610 − 0.610i)27-s + (−1.19 − 1.19i)29-s − 0.109·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.728i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.685 + 0.728i$
Analytic conductor: \(18.8806\)
Root analytic conductor: \(4.34518\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :3/2),\ -0.685 + 0.728i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4061375669\)
\(L(\frac12)\) \(\approx\) \(0.4061375669\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.53 - 3.53i)T \)
good3 \( 1 + (-1.80 - 1.80i)T + 27iT^{2} \)
7 \( 1 - 5.36iT - 343T^{2} \)
11 \( 1 + (26.5 - 26.5i)T - 1.33e3iT^{2} \)
13 \( 1 + (45.3 + 45.3i)T + 2.19e3iT^{2} \)
17 \( 1 + 11.5T + 4.91e3T^{2} \)
19 \( 1 + (-13.0 - 13.0i)T + 6.85e3iT^{2} \)
23 \( 1 + 33.7iT - 1.21e4T^{2} \)
29 \( 1 + (187. + 187. i)T + 2.43e4iT^{2} \)
31 \( 1 + 18.8T + 2.97e4T^{2} \)
37 \( 1 + (191. - 191. i)T - 5.06e4iT^{2} \)
41 \( 1 + 380. iT - 6.89e4T^{2} \)
43 \( 1 + (-140. + 140. i)T - 7.95e4iT^{2} \)
47 \( 1 + 399.T + 1.03e5T^{2} \)
53 \( 1 + (92.8 - 92.8i)T - 1.48e5iT^{2} \)
59 \( 1 + (463. - 463. i)T - 2.05e5iT^{2} \)
61 \( 1 + (96.3 + 96.3i)T + 2.26e5iT^{2} \)
67 \( 1 + (329. + 329. i)T + 3.00e5iT^{2} \)
71 \( 1 + 654. iT - 3.57e5T^{2} \)
73 \( 1 + 147. iT - 3.89e5T^{2} \)
79 \( 1 + 1.07e3T + 4.93e5T^{2} \)
83 \( 1 + (-866. - 866. i)T + 5.71e5iT^{2} \)
89 \( 1 + 452. iT - 7.04e5T^{2} \)
97 \( 1 - 649.T + 9.12e5T^{2} \)
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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

−10.65247595297508541063105104864, −9.952323369972743250881001892481, −9.082703729070651716323513612941, −7.935256250591427702254081123700, −7.16375950863107207024674007907, −5.84372090111565182009178164873, −4.69803049378719963149272201558, −3.45350751508078703706422180079, −2.35747457198138378938207827461, −0.13171121666591910646949200593, 1.73437754035899053990875528616, 3.07073716102203339776999861635, 4.53097420868365610774050230975, 5.48027414925018819392681813175, 7.01406363857513755322211303417, 7.69554703085349928799361955271, 8.626191922161534467519872084807, 9.579362250345813229903541581352, 10.76631657617421071765269570635, 11.44973302400024814506729680457

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