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Properties

Label 2-160-160.107-c3-0-35
Degree $2$
Conductor $160$
Sign $0.992 - 0.122i$
Analytic cond. $9.44030$
Root an. cond. $3.07250$
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  + (−2.18 − 1.79i)2-s + (7.10 + 2.94i)3-s + (1.55 + 7.84i)4-s + (10.5 + 3.62i)5-s + (−10.2 − 19.1i)6-s + 2.95·7-s + (10.6 − 19.9i)8-s + (22.7 + 22.7i)9-s + (−16.6 − 26.9i)10-s + (15.3 − 37.1i)11-s + (−12.0 + 60.3i)12-s + (15.7 + 6.51i)13-s + (−6.46 − 5.31i)14-s + (64.4 + 56.8i)15-s + (−59.1 + 24.3i)16-s + (−34.2 + 34.2i)17-s + ⋯
L(s)  = 1  + (−0.772 − 0.634i)2-s + (1.36 + 0.566i)3-s + (0.194 + 0.980i)4-s + (0.945 + 0.324i)5-s + (−0.696 − 1.30i)6-s + 0.159·7-s + (0.472 − 0.881i)8-s + (0.840 + 0.840i)9-s + (−0.524 − 0.851i)10-s + (0.421 − 1.01i)11-s + (−0.289 + 1.45i)12-s + (0.335 + 0.139i)13-s + (−0.123 − 0.101i)14-s + (1.10 + 0.979i)15-s + (−0.924 + 0.380i)16-s + (−0.488 + 0.488i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.992 - 0.122i$
Analytic conductor: \(9.44030\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :3/2),\ 0.992 - 0.122i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.17439 + 0.133486i\)
\(L(\frac12)\) \(\approx\) \(2.17439 + 0.133486i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.18 + 1.79i)T \)
5 \( 1 + (-10.5 - 3.62i)T \)
good3 \( 1 + (-7.10 - 2.94i)T + (19.0 + 19.0i)T^{2} \)
7 \( 1 - 2.95T + 343T^{2} \)
11 \( 1 + (-15.3 + 37.1i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (-15.7 - 6.51i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + (34.2 - 34.2i)T - 4.91e3iT^{2} \)
19 \( 1 + (-31.8 - 76.8i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + 38.1T + 1.21e4T^{2} \)
29 \( 1 + (30.6 + 74.0i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 122. iT - 2.97e4T^{2} \)
37 \( 1 + (-22.0 + 9.12i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-325. + 325. i)T - 6.89e4iT^{2} \)
43 \( 1 + (147. + 355. i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + (-296. - 296. i)T + 1.03e5iT^{2} \)
53 \( 1 + (287. - 119. i)T + (1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (177. - 428. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (830. - 344. i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (-285. + 689. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (48.8 - 48.8i)T - 3.57e5iT^{2} \)
73 \( 1 + 603. iT - 3.89e5T^{2} \)
79 \( 1 - 642. iT - 4.93e5T^{2} \)
83 \( 1 + (-366. + 885. i)T + (-4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (772. - 772. i)T - 7.04e5iT^{2} \)
97 \( 1 + (916. + 916. i)T + 9.12e5iT^{2} \)
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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

−12.43009541431784830208833578624, −11.02138056064117159782199468001, −10.26332874123941839522929008278, −9.253946391064849652330232014627, −8.736708730281610785155771446329, −7.68698625530000614991067491602, −6.12656002287468258513865185272, −3.97213586142730126323823778687, −2.96722537148488824238324950816, −1.70568888773887357586493383748, 1.41505246814984426864573464249, 2.50951188563612061627458830999, 4.80320813610579441071706486529, 6.36035467173971117404496675121, 7.33809802192082752014092790095, 8.309745862738522147496787128066, 9.309648554331186552391240149360, 9.684757967413166762450712148866, 11.19923274913198922617044622995, 12.81534157079308329259672872523

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