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Properties

Label 2-342-19.11-c3-0-23
Degree $2$
Conductor $342$
Sign $-0.658 - 0.752i$
Analytic cond. $20.1786$
Root an. cond. $4.49206$
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 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (2.09 − 3.62i)5-s + 3.18·7-s − 7.99·8-s + (−4.18 − 7.25i)10-s − 69.4·11-s + (4.06 + 7.03i)13-s + (3.18 − 5.52i)14-s + (−8 + 13.8i)16-s + (−53.0 + 91.8i)17-s + (42.6 − 70.9i)19-s − 16.7·20-s + (−69.4 + 120. i)22-s + (−88.2 − 152. i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.187 − 0.324i)5-s + 0.172·7-s − 0.353·8-s + (−0.132 − 0.229i)10-s − 1.90·11-s + (0.0867 + 0.150i)13-s + (0.0608 − 0.105i)14-s + (−0.125 + 0.216i)16-s + (−0.756 + 1.31i)17-s + (0.515 − 0.857i)19-s − 0.187·20-s + (−0.672 + 1.16i)22-s + (−0.800 − 1.38i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.658 - 0.752i$
Analytic conductor: \(20.1786\)
Root analytic conductor: \(4.49206\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :3/2),\ -0.658 - 0.752i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1796972533\)
\(L(\frac12)\) \(\approx\) \(0.1796972533\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
19 \( 1 + (-42.6 + 70.9i)T \)
good5 \( 1 + (-2.09 + 3.62i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 - 3.18T + 343T^{2} \)
11 \( 1 + 69.4T + 1.33e3T^{2} \)
13 \( 1 + (-4.06 - 7.03i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (53.0 - 91.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (88.2 + 152. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (33.1 + 57.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 140.T + 2.97e4T^{2} \)
37 \( 1 + 156.T + 5.06e4T^{2} \)
41 \( 1 + (207. - 359. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (57.9 - 100. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-310. - 537. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (185. + 322. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-45.8 + 79.4i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (109. + 189. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-72.6 - 125. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-443. + 768. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (99.5 - 172. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-194. + 337. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 380.T + 5.71e5T^{2} \)
89 \( 1 + (212. + 368. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (209. - 363. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.70138891939907576613352734637, −9.790349375866149488933067003872, −8.670369306494728716451130119572, −7.83555697025880868609347169445, −6.41043967445246368256443448616, −5.28492461426173211446214832597, −4.48391680781548317038092242385, −2.99669298242396030969692466522, −1.84650258791015523442720045075, −0.05162664695882702425637287273, 2.31200060207051063190086304675, 3.54769411364595463017613845892, 5.08803998349905132002033947847, 5.62036726182807570162989298031, 7.04400744167764005726699015982, 7.69557121719748655253220429301, 8.689088912628561285645119815666, 9.895541374035683206358440601738, 10.69301756352104108770944625102, 11.76917299056238083642740803756

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