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Properties

Label 2-342-19.11-c5-0-13
Degree $2$
Conductor $342$
Sign $0.178 - 0.983i$
Analytic cond. $54.8512$
Root an. cond. $7.40616$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 3.46i)2-s + (−7.99 − 13.8i)4-s + (−34.6 + 59.9i)5-s − 195.·7-s + 63.9·8-s + (−138. − 239. i)10-s + 137.·11-s + (−398. − 689. i)13-s + (390. − 676. i)14-s + (−128 + 221. i)16-s + (784. − 1.35e3i)17-s + (1.55e3 − 206. i)19-s + 1.10e3·20-s + (−274. + 476. i)22-s + (−208. − 361. i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.619 + 1.07i)5-s − 1.50·7-s + 0.353·8-s + (−0.437 − 0.758i)10-s + 0.342·11-s + (−0.653 − 1.13i)13-s + (0.532 − 0.922i)14-s + (−0.125 + 0.216i)16-s + (0.658 − 1.14i)17-s + (0.991 − 0.131i)19-s + 0.619·20-s + (−0.121 + 0.209i)22-s + (−0.0822 − 0.142i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.178 - 0.983i$
Analytic conductor: \(54.8512\)
Root analytic conductor: \(7.40616\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :5/2),\ 0.178 - 0.983i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7751477541\)
\(L(\frac12)\) \(\approx\) \(0.7751477541\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 - 3.46i)T \)
3 \( 1 \)
19 \( 1 + (-1.55e3 + 206. i)T \)
good5 \( 1 + (34.6 - 59.9i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + 195.T + 1.68e4T^{2} \)
11 \( 1 - 137.T + 1.61e5T^{2} \)
13 \( 1 + (398. + 689. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-784. + 1.35e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
23 \( 1 + (208. + 361. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (155. + 270. i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + 1.03e4T + 2.86e7T^{2} \)
37 \( 1 - 3.32e3T + 6.93e7T^{2} \)
41 \( 1 + (2.54e3 - 4.39e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (4.08e3 - 7.08e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-1.01e3 - 1.76e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-6.07e3 - 1.05e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (2.36e3 - 4.09e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (4.31e3 + 7.46e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (6.73e3 + 1.16e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (3.50e4 - 6.07e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (-9.04e3 + 1.56e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-4.40e4 + 7.62e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 1.15e5T + 3.93e9T^{2} \)
89 \( 1 + (-1.29e4 - 2.24e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + (6.50e4 - 1.12e5i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−10.68296474310617743819869986453, −9.831178895800738939392510715271, −9.229458662249739198460658982856, −7.64613586181612336157896084838, −7.25445507410177248496439836535, −6.29586569584496994913196879148, −5.24767336821074943324129170346, −3.52484600355194843549637978329, −2.86928156431347748425530959534, −0.57675825619641987238508673830, 0.43211018797806734737815996635, 1.74061887459757789117814219567, 3.38755294729733599398665473240, 4.10870610375781083382642696813, 5.46080301855572447178234242565, 6.78884810030814128673057728076, 7.79616521358481498498353648182, 8.981951939197061819381459067616, 9.425217452986743639995706276942, 10.33643084828134895063968390552

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