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Properties

Label 2-161-7.2-c3-0-24
Degree $2$
Conductor $161$
Sign $0.880 + 0.474i$
Analytic cond. $9.49930$
Root an. cond. $3.08209$
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  + (−1.82 + 3.16i)2-s + (−0.0618 − 0.107i)3-s + (−2.68 − 4.65i)4-s + (4.27 − 7.39i)5-s + 0.452·6-s + (−9.80 + 15.7i)7-s − 9.60·8-s + (13.4 − 23.3i)9-s + (15.6 + 27.0i)10-s + (−14.5 − 25.2i)11-s + (−0.332 + 0.575i)12-s − 0.185·13-s + (−31.8 − 59.7i)14-s − 1.05·15-s + (39.0 − 67.6i)16-s + (−57.1 − 98.9i)17-s + ⋯
L(s)  = 1  + (−0.646 + 1.11i)2-s + (−0.0118 − 0.0206i)3-s + (−0.335 − 0.581i)4-s + (0.382 − 0.661i)5-s + 0.0307·6-s + (−0.529 + 0.848i)7-s − 0.424·8-s + (0.499 − 0.865i)9-s + (0.493 + 0.855i)10-s + (−0.399 − 0.692i)11-s + (−0.00798 + 0.0138i)12-s − 0.00395·13-s + (−0.607 − 1.14i)14-s − 0.0181·15-s + (0.610 − 1.05i)16-s + (−0.815 − 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.880 + 0.474i$
Analytic conductor: \(9.49930\)
Root analytic conductor: \(3.08209\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :3/2),\ 0.880 + 0.474i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.793942 - 0.200472i\)
\(L(\frac12)\) \(\approx\) \(0.793942 - 0.200472i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (9.80 - 15.7i)T \)
23 \( 1 + (-11.5 + 19.9i)T \)
good2 \( 1 + (1.82 - 3.16i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (0.0618 + 0.107i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-4.27 + 7.39i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (14.5 + 25.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 0.185T + 2.19e3T^{2} \)
17 \( 1 + (57.1 + 98.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-16.7 + 29.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
29 \( 1 - 227.T + 2.43e4T^{2} \)
31 \( 1 + (86.9 + 150. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (1.42 - 2.47i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 462.T + 6.89e4T^{2} \)
43 \( 1 + 366.T + 7.95e4T^{2} \)
47 \( 1 + (87.8 - 152. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (214. + 372. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (274. + 475. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-193. + 335. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (51.2 + 88.7i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 350.T + 3.57e5T^{2} \)
73 \( 1 + (313. + 543. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (547. - 947. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 8.85T + 5.71e5T^{2} \)
89 \( 1 + (565. - 980. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 283.T + 9.12e5T^{2} \)
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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

−12.45555417783204971520001513602, −11.42223078274879790527189657443, −9.606008578564895042654671589048, −9.210066833954021378734650038094, −8.290136456427338286980548295677, −6.95394995598592492135203777771, −6.12800280273934480212137147093, −5.03080195919775029803268703818, −2.93069501220081565774370798505, −0.47945572496593761851928151969, 1.57951371054194173245268127821, 2.83637619765831652730070718418, 4.32954414077761316437772331396, 6.20682942676014516188044785857, 7.32648305226615235168201003025, 8.637926943402559363679221068298, 10.11270213267912541272958141679, 10.27932974985143735270934981228, 11.05481139862375652050452700533, 12.46179410042536945130980106111

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