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Properties

Label 2-161-7.2-c3-0-23
Degree $2$
Conductor $161$
Sign $0.537 - 0.842i$
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  + (0.674 − 1.16i)2-s + (3.89 + 6.74i)3-s + (3.08 + 5.35i)4-s + (3.28 − 5.68i)5-s + 10.5·6-s + (14.9 + 10.9i)7-s + 19.1·8-s + (−16.8 + 29.1i)9-s + (−4.43 − 7.67i)10-s + (−20.1 − 34.9i)11-s + (−24.0 + 41.6i)12-s + 33.8·13-s + (22.8 − 10.0i)14-s + 51.1·15-s + (−11.8 + 20.4i)16-s + (−48.0 − 83.3i)17-s + ⋯
L(s)  = 1  + (0.238 − 0.413i)2-s + (0.749 + 1.29i)3-s + (0.386 + 0.668i)4-s + (0.293 − 0.508i)5-s + 0.714·6-s + (0.807 + 0.590i)7-s + 0.845·8-s + (−0.622 + 1.07i)9-s + (−0.140 − 0.242i)10-s + (−0.553 − 0.958i)11-s + (−0.578 + 1.00i)12-s + 0.721·13-s + (0.436 − 0.192i)14-s + 0.880·15-s + (−0.184 + 0.319i)16-s + (−0.686 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.842i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.537 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.537 - 0.842i$
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: \((2,\ 161,\ (\ :3/2),\ 0.537 - 0.842i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.59334 + 1.42152i\)
\(L(\frac12)\) \(\approx\) \(2.59334 + 1.42152i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-14.9 - 10.9i)T \)
23 \( 1 + (-11.5 + 19.9i)T \)
good2 \( 1 + (-0.674 + 1.16i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-3.89 - 6.74i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-3.28 + 5.68i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (20.1 + 34.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 33.8T + 2.19e3T^{2} \)
17 \( 1 + (48.0 + 83.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (60.7 - 105. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
29 \( 1 + 173.T + 2.43e4T^{2} \)
31 \( 1 + (108. + 188. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-83.5 + 144. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 12.2T + 6.89e4T^{2} \)
43 \( 1 - 463.T + 7.95e4T^{2} \)
47 \( 1 + (-25.9 + 45.0i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (340. + 589. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (156. + 270. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (176. - 304. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (380. + 659. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 231.T + 3.57e5T^{2} \)
73 \( 1 + (-354. - 614. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-483. + 837. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 185.T + 5.71e5T^{2} \)
89 \( 1 + (684. - 1.18e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 419.T + 9.12e5T^{2} \)
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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

−12.62893827598609591816844085109, −11.23239648577201643621145573720, −10.86568849117051752539853675125, −9.369346277633209413094264285716, −8.632678308110454303923624323988, −7.78905709604838881107734367356, −5.69735471433146362475785506581, −4.49882315071961843510796469210, −3.44812805233385238507138618250, −2.18145381692967551969338183233, 1.43869400251047523563165211544, 2.37680812775363862851903572523, 4.54584784771482425131242366736, 6.15313956615447258019189440101, 7.02475044992204292254137600843, 7.69428781587778136693114740890, 8.880999709977507561166345519053, 10.53377557959153544032607472821, 11.04081172414880113948204978023, 12.68069785863097005388254797846

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