Properties

Label 2-342-19.11-c1-0-3
Degree $2$
Conductor $342$
Sign $0.761 + 0.648i$
Analytic cond. $2.73088$
Root an. cond. $1.65253$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.822 + 1.42i)5-s + 3.64·7-s − 0.999·8-s + (0.822 + 1.42i)10-s + 4.64·11-s + (−1 − 1.73i)13-s + (1.82 − 3.15i)14-s + (−0.5 + 0.866i)16-s + (1.67 − 4.02i)19-s + 1.64·20-s + (2.32 − 4.02i)22-s + (−0.822 − 1.42i)23-s + (1.14 + 1.98i)25-s − 1.99·26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.368 + 0.637i)5-s + 1.37·7-s − 0.353·8-s + (0.260 + 0.450i)10-s + 1.40·11-s + (−0.277 − 0.480i)13-s + (0.487 − 0.843i)14-s + (−0.125 + 0.216i)16-s + (0.384 − 0.923i)19-s + 0.368·20-s + (0.495 − 0.857i)22-s + (−0.171 − 0.297i)23-s + (0.229 + 0.396i)25-s − 0.392·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.761 + 0.648i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.761 + 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60503 - 0.591064i\)
\(L(\frac12)\) \(\approx\) \(1.60503 - 0.591064i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
19 \( 1 + (-1.67 + 4.02i)T \)
good5 \( 1 + (0.822 - 1.42i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.64T + 7T^{2} \)
11 \( 1 - 4.64T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.822 + 1.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.822 - 1.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.64T + 31T^{2} \)
37 \( 1 - 0.354T + 37T^{2} \)
41 \( 1 + (0.145 - 0.252i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.64 - 9.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.17 - 3.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.29 + 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.96 - 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.468 + 0.811i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.322 + 0.559i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.35 - 2.34i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.854 - 1.47i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.93T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.85 + 3.21i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.31159884404933973600836681940, −10.94493856837592531855013355133, −9.685616491286377136968104455513, −8.712493465710741979693879563308, −7.61374211172834086920546577787, −6.60559310288547236986515384968, −5.22162081567605308473383437648, −4.26828508120732633171732882489, −3.05656070384125913009178148614, −1.50049463273171998562474178605, 1.60365219258448949592715104002, 3.83696549258920490638066709528, 4.63027419333522932580151864152, 5.63734521930319899989259579684, 6.88639474737511592823011646712, 7.87722227169837887625263063753, 8.626253213404958961491213169549, 9.484851511018492196567236500406, 10.93516864845477266092419506321, 11.99632033772289023759836888510

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