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Properties

Label 2-342-19.7-c3-0-13
Degree $2$
Conductor $342$
Sign $-0.0661 - 0.997i$
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 + (7.12 + 12.3i)5-s + 13.2·7-s − 7.99·8-s + (−14.2 + 24.6i)10-s + 65.9·11-s + (34.2 − 59.3i)13-s + (13.2 + 22.9i)14-s + (−8 − 13.8i)16-s + (49.8 + 86.3i)17-s + (−80.3 − 20.0i)19-s − 57.0·20-s + (65.9 + 114. i)22-s + (1.76 − 3.06i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.637 + 1.10i)5-s + 0.715·7-s − 0.353·8-s + (−0.450 + 0.780i)10-s + 1.80·11-s + (0.730 − 1.26i)13-s + (0.253 + 0.438i)14-s + (−0.125 − 0.216i)16-s + (0.711 + 1.23i)17-s + (−0.970 − 0.242i)19-s − 0.637·20-s + (0.638 + 1.10i)22-s + (0.0160 − 0.0277i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0661 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0661 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.0661 - 0.997i$
Analytic conductor: \(20.1786\)
Root analytic conductor: \(4.49206\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :3/2),\ -0.0661 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.956390936\)
\(L(\frac12)\) \(\approx\) \(2.956390936\)
\(L(\frac{5}{2})\) 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 \( 1 + (80.3 + 20.0i)T \)
good5 \( 1 + (-7.12 - 12.3i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 - 13.2T + 343T^{2} \)
11 \( 1 - 65.9T + 1.33e3T^{2} \)
13 \( 1 + (-34.2 + 59.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-49.8 - 86.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-1.76 + 3.06i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (40.9 - 70.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 247.T + 2.97e4T^{2} \)
37 \( 1 - 421.T + 5.06e4T^{2} \)
41 \( 1 + (172. + 299. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-183. - 317. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (45.4 - 78.7i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (344. - 596. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-91.1 - 157. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-0.258 + 0.448i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-79.9 + 138. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (395. + 685. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (161. + 278. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (159. + 275. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 684.T + 5.71e5T^{2} \)
89 \( 1 + (-360. + 623. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (207. + 359. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.13728390719186645753582234397, −10.61970772117055058585330590073, −9.379964846101570506296608737577, −8.390146264297239110652895075902, −7.40603201582326276317533906933, −6.25768819521141167411392636267, −5.87096799071996685483120420569, −4.24460409699974910187907389699, −3.21967327701121583557809594693, −1.55485856858961218961363597163, 1.12729544927684568452641432628, 1.88915954004209040388482609758, 3.86717035981267289443873215206, 4.65072695606462051324532141314, 5.74764205030528362630012065056, 6.78561506105276278752142670203, 8.387550592853367119879876482510, 9.234207051001046291634491043871, 9.664651663034179845640787868666, 11.32146823780836911810100508103

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