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Properties

Label 2-342-19.6-c1-0-3
Degree $2$
Conductor $342$
Sign $0.988 - 0.151i$
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  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.347 − 1.96i)5-s + (−1.34 + 2.33i)7-s + (0.500 + 0.866i)8-s + (1.53 + 1.28i)10-s + (1.59 + 2.75i)11-s + (5.41 − 1.96i)13-s + (−0.467 − 2.65i)14-s + (−0.939 − 0.342i)16-s + (4.99 − 4.18i)17-s + (2.82 − 3.31i)19-s − 2·20-s + (−2.99 − 1.08i)22-s + (0.120 − 0.684i)23-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (0.0868 − 0.492i)4-s + (−0.155 − 0.880i)5-s + (−0.509 + 0.882i)7-s + (0.176 + 0.306i)8-s + (0.484 + 0.406i)10-s + (0.480 + 0.831i)11-s + (1.50 − 0.546i)13-s + (−0.125 − 0.709i)14-s + (−0.234 − 0.0855i)16-s + (1.21 − 1.01i)17-s + (0.648 − 0.761i)19-s − 0.447·20-s + (−0.638 − 0.232i)22-s + (0.0251 − 0.142i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.988 - 0.151i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.988 - 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03794 + 0.0790452i\)
\(L(\frac12)\) \(\approx\) \(1.03794 + 0.0790452i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 \)
19 \( 1 + (-2.82 + 3.31i)T \)
good5 \( 1 + (0.347 + 1.96i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (1.34 - 2.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.59 - 2.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.41 + 1.96i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-4.99 + 4.18i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.120 + 0.684i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.16 - 1.81i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.22 - 2.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.36T + 37T^{2} \)
41 \( 1 + (0.326 + 0.118i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.05 - 5.97i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (6.04 + 5.06i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.42 - 8.08i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (0.439 - 0.368i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.509 + 2.89i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (3.79 + 3.18i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.46 - 8.32i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (14.8 + 5.39i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-8.51 - 3.10i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-4.23 + 7.34i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.27 + 2.64i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (0.266 - 0.223i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.66388159767511034552580632910, −10.41666633521128953322791512472, −9.342573960870860147408032163783, −8.901538111388857042873367440814, −7.910511469947537965241693611865, −6.79056850681255658698294938007, −5.70860738690751212365143539343, −4.82554153840372801171162356188, −3.13871059777718496909182580921, −1.16710136654501543480516522240, 1.29748251665886560014252114111, 3.40497838904513841759386813870, 3.74496776699232691304905958972, 5.94305387016245168812577894652, 6.76341586719595585848317145606, 7.82004211595146683811710473352, 8.724861417891740634502655138962, 9.895477043791285110598702860712, 10.59671898487992377840514131885, 11.25288368290351490669965067138

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