↑

Properties

Label 2-336-112.109-c1-0-10
Degree $2$
Conductor $336$
Sign $0.364 - 0.931i$
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  + (1.25 + 0.646i)2-s + (−0.965 + 0.258i)3-s + (1.16 + 1.62i)4-s + (−0.162 − 0.0435i)5-s + (−1.38 − 0.299i)6-s + (2.42 − 1.06i)7-s + (0.410 + 2.79i)8-s + (0.866 − 0.499i)9-s + (−0.176 − 0.160i)10-s + (1.33 + 4.98i)11-s + (−1.54 − 1.27i)12-s + (−1.17 − 1.17i)13-s + (3.73 + 0.232i)14-s + 0.168·15-s + (−1.29 + 3.78i)16-s + (−0.638 + 1.10i)17-s + ⋯
L(s)  = 1  + (0.889 + 0.457i)2-s + (−0.557 + 0.149i)3-s + (0.581 + 0.813i)4-s + (−0.0727 − 0.0194i)5-s + (−0.564 − 0.122i)6-s + (0.916 − 0.401i)7-s + (0.145 + 0.989i)8-s + (0.288 − 0.166i)9-s + (−0.0557 − 0.0506i)10-s + (0.402 + 1.50i)11-s + (−0.445 − 0.366i)12-s + (−0.326 − 0.326i)13-s + (0.998 + 0.0622i)14-s + 0.0434·15-s + (−0.323 + 0.946i)16-s + (−0.154 + 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.364 - 0.931i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.364 - 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63349 + 1.11538i\)
\(L(\frac12)\) \(\approx\) \(1.63349 + 1.11538i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 - 0.646i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (-2.42 + 1.06i)T \)
good5 \( 1 + (0.162 + 0.0435i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.33 - 4.98i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.17 + 1.17i)T + 13iT^{2} \)
17 \( 1 + (0.638 - 1.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0895 - 0.334i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.67 + 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.533 - 0.533i)T + 29iT^{2} \)
31 \( 1 + (-1.39 + 2.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.57 + 1.49i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 9.90iT - 41T^{2} \)
43 \( 1 + (-6.90 + 6.90i)T - 43iT^{2} \)
47 \( 1 + (0.264 + 0.457i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.369 + 1.37i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.10 - 11.6i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.83 + 10.5i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (8.19 - 2.19i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.77iT - 71T^{2} \)
73 \( 1 + (10.4 + 6.04i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.73 - 6.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.12 + 8.12i)T + 83iT^{2} \)
89 \( 1 + (9.53 - 5.50i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.51T + 97T^{2} \)
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

−12.03361070650315492203828095121, −10.97162827115589032257404656829, −10.16842258574533174970277213280, −8.738103503849666541417730026072, −7.52249565088055760142645071704, −6.96583066987591328189466857352, −5.66946524064025743893616521058, −4.70957113227787799945805568647, −4.00223377597971104002523815248, −2.07294287895466385736161566528, 1.37281132480542361848367991691, 2.99616014980114474365155401009, 4.38896342485821675305015244421, 5.38256502057346742526933161935, 6.17399737159135953558887163773, 7.31937988188006751180920663686, 8.599685022175988380345584977044, 9.747914305195825150157253434458, 11.07289826681386328109840129060, 11.34662739307378205644196576603

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