↑

Properties

Label 2-336-48.11-c1-0-7
Degree $2$
Conductor $336$
Sign $0.986 + 0.162i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.230 − 1.39i)2-s + (−1.57 − 0.714i)3-s + (−1.89 + 0.642i)4-s + (−1.31 + 1.31i)5-s + (−0.633 + 2.36i)6-s − 7-s + (1.33 + 2.49i)8-s + (1.97 + 2.25i)9-s + (2.14 + 1.53i)10-s + (1.23 + 1.23i)11-s + (3.44 + 0.339i)12-s + (4.57 − 4.57i)13-s + (0.230 + 1.39i)14-s + (3.02 − 1.13i)15-s + (3.17 − 2.43i)16-s + 5.82i·17-s + ⋯
L(s)  = 1  + (−0.162 − 0.986i)2-s + (−0.910 − 0.412i)3-s + (−0.946 + 0.321i)4-s + (−0.590 + 0.590i)5-s + (−0.258 + 0.965i)6-s − 0.377·7-s + (0.471 + 0.881i)8-s + (0.659 + 0.751i)9-s + (0.678 + 0.486i)10-s + (0.372 + 0.372i)11-s + (0.995 + 0.0980i)12-s + (1.26 − 1.26i)13-s + (0.0615 + 0.372i)14-s + (0.781 − 0.294i)15-s + (0.793 − 0.608i)16-s + 1.41i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.986 + 0.162i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.986 + 0.162i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.660688 - 0.0542027i\)
\(L(\frac12)\) \(\approx\) \(0.660688 - 0.0542027i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.230 + 1.39i)T \)
3 \( 1 + (1.57 + 0.714i)T \)
7 \( 1 + T \)
good5 \( 1 + (1.31 - 1.31i)T - 5iT^{2} \)
11 \( 1 + (-1.23 - 1.23i)T + 11iT^{2} \)
13 \( 1 + (-4.57 + 4.57i)T - 13iT^{2} \)
17 \( 1 - 5.82iT - 17T^{2} \)
19 \( 1 + (2.27 + 2.27i)T + 19iT^{2} \)
23 \( 1 - 2.63iT - 23T^{2} \)
29 \( 1 + (-4.30 - 4.30i)T + 29iT^{2} \)
31 \( 1 - 5.81iT - 31T^{2} \)
37 \( 1 + (-6.99 - 6.99i)T + 37iT^{2} \)
41 \( 1 + 3.57T + 41T^{2} \)
43 \( 1 + (-2.12 + 2.12i)T - 43iT^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + (-0.573 + 0.573i)T - 53iT^{2} \)
59 \( 1 + (8.42 + 8.42i)T + 59iT^{2} \)
61 \( 1 + (6.80 - 6.80i)T - 61iT^{2} \)
67 \( 1 + (-1.90 - 1.90i)T + 67iT^{2} \)
71 \( 1 - 13.3iT - 71T^{2} \)
73 \( 1 - 0.516iT - 73T^{2} \)
79 \( 1 + 3.10iT - 79T^{2} \)
83 \( 1 + (2.91 - 2.91i)T - 83iT^{2} \)
89 \( 1 + 4.34T + 89T^{2} \)
97 \( 1 - 8.09T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.38131833228816587646639251632, −10.71540345123098245919199177341, −10.22223931995770464259573203680, −8.742129466913387879016616876170, −7.84030323509538817852302320271, −6.67445014896641927367196480446, −5.59718812073998095560225760581, −4.20308736278263089236339251167, −3.13351951796307799645670738893, −1.28346498526628610303512037435, 0.67275426419775752308974904508, 4.00778210961664354107525818195, 4.53586874528180885705005571239, 5.95734411006313193172117669024, 6.49202112643227329728710947905, 7.69291619284228220987335730810, 8.897251163004361707093344135714, 9.410742249671432511233831277142, 10.62648032787178943828390830910, 11.61428411673714197667675858450

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