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Properties

Label 2-342-19.11-c3-0-15
Degree $2$
Conductor $342$
Sign $0.281 + 0.959i$
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 + (4.68 − 8.11i)5-s + 29.9·7-s − 7.99·8-s + (−9.36 − 16.2i)10-s + 38.2·11-s + (21.8 + 37.9i)13-s + (29.9 − 51.9i)14-s + (−8 + 13.8i)16-s + (−40.9 + 70.8i)17-s + (68.5 − 46.4i)19-s − 37.4·20-s + (38.2 − 66.2i)22-s + (−32.2 − 55.9i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.418 − 0.725i)5-s + 1.61·7-s − 0.353·8-s + (−0.296 − 0.513i)10-s + 1.04·11-s + (0.467 + 0.809i)13-s + (0.572 − 0.991i)14-s + (−0.125 + 0.216i)16-s + (−0.583 + 1.01i)17-s + (0.827 − 0.561i)19-s − 0.418·20-s + (0.370 − 0.642i)22-s + (−0.292 − 0.506i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.281 + 0.959i$
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: \((2,\ 342,\ (\ :3/2),\ 0.281 + 0.959i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.022720869\)
\(L(\frac12)\) \(\approx\) \(3.022720869\)
\(L(\frac{5}{2})\) 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 \( 1 + (-68.5 + 46.4i)T \)
good5 \( 1 + (-4.68 + 8.11i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 - 29.9T + 343T^{2} \)
11 \( 1 - 38.2T + 1.33e3T^{2} \)
13 \( 1 + (-21.8 - 37.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (40.9 - 70.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (32.2 + 55.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (33.8 + 58.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 142.T + 2.97e4T^{2} \)
37 \( 1 + 345.T + 5.06e4T^{2} \)
41 \( 1 + (-242. + 419. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (77.5 - 134. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (222. + 384. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (14.7 + 25.6i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-158. + 275. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-146. - 252. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (197. + 342. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (357. - 618. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (583. - 1.01e3i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (111. - 192. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 601.T + 5.71e5T^{2} \)
89 \( 1 + (694. + 1.20e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (80.2 - 138. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.23592459656074138188250404758, −10.09647533669792916367761657617, −8.879653880665607293538131021905, −8.534095835783299739800518378747, −6.97852607586265659179975018497, −5.70928389948092135586851453616, −4.72812958139695097664366229162, −3.94362004645920034734501489113, −1.96843681193540827654397387974, −1.22223933069276044486582383040, 1.43223570416662452165974874566, 3.04792981582120552302930602357, 4.43011554421061593968892904989, 5.41327093180764619897080417538, 6.43283767901921666763488774419, 7.44587344524652427785299052395, 8.259134908367587425527152271370, 9.288143868383581697984581111962, 10.46520018428397695744979903912, 11.41923245615561210846489144358

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