Properties

Label 2-3328-8.5-c1-0-93
Degree 22
Conductor 33283328
Sign 0.7070.707i-0.707 - 0.707i
Analytic cond. 26.574226.5742
Root an. cond. 5.155015.15501
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.23i·3-s − 3i·5-s + 2.23·7-s − 2.00·9-s − 4.47i·11-s i·13-s − 6.70·15-s − 3·17-s + 4.47i·19-s − 5.00i·21-s − 8.94·23-s − 4·25-s − 2.23i·27-s + 10i·29-s − 10.0·33-s + ⋯
L(s)  = 1  − 1.29i·3-s − 1.34i·5-s + 0.845·7-s − 0.666·9-s − 1.34i·11-s − 0.277i·13-s − 1.73·15-s − 0.727·17-s + 1.02i·19-s − 1.09i·21-s − 1.86·23-s − 0.800·25-s − 0.430i·27-s + 1.85i·29-s − 1.74·33-s + ⋯

Functional equation

Λ(s)=(3328s/2ΓC(s)L(s)=((0.7070.707i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3328s/2ΓC(s+1/2)L(s)=((0.7070.707i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33283328    =    28132^{8} \cdot 13
Sign: 0.7070.707i-0.707 - 0.707i
Analytic conductor: 26.574226.5742
Root analytic conductor: 5.155015.15501
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3328(1665,)\chi_{3328} (1665, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3328, ( :1/2), 0.7070.707i)(2,\ 3328,\ (\ :1/2),\ -0.707 - 0.707i)

Particular Values

L(1)L(1) \approx 1.2655253161.265525316
L(12)L(\frac12) \approx 1.2655253161.265525316
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
13 1+iT 1 + iT
good3 1+2.23iT3T2 1 + 2.23iT - 3T^{2}
5 1+3iT5T2 1 + 3iT - 5T^{2}
7 12.23T+7T2 1 - 2.23T + 7T^{2}
11 1+4.47iT11T2 1 + 4.47iT - 11T^{2}
17 1+3T+17T2 1 + 3T + 17T^{2}
19 14.47iT19T2 1 - 4.47iT - 19T^{2}
23 1+8.94T+23T2 1 + 8.94T + 23T^{2}
29 110iT29T2 1 - 10iT - 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+3iT37T2 1 + 3iT - 37T^{2}
41 1+41T2 1 + 41T^{2}
43 16.70iT43T2 1 - 6.70iT - 43T^{2}
47 1+2.23T+47T2 1 + 2.23T + 47T^{2}
53 1+4iT53T2 1 + 4iT - 53T^{2}
59 1+4.47iT59T2 1 + 4.47iT - 59T^{2}
61 161T2 1 - 61T^{2}
67 1+13.4iT67T2 1 + 13.4iT - 67T^{2}
71 16.70T+71T2 1 - 6.70T + 71T^{2}
73 1+14T+73T2 1 + 14T + 73T^{2}
79 18.94T+79T2 1 - 8.94T + 79T^{2}
83 1+17.8iT83T2 1 + 17.8iT - 83T^{2}
89 110T+89T2 1 - 10T + 89T^{2}
97 1+2T+97T2 1 + 2T + 97T^{2}
show more
show less
   L(s)=p j=12(1αj,pps)1L(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

−8.068793140220475787863403631665, −7.78171600246497402485886233207, −6.58729584588435536777439207868, −5.94253520243105416126131206293, −5.23116982894672747637051606040, −4.39125785254593678461574087875, −3.35588060359060981566985045467, −1.92462084059525295215857969172, −1.41432921684726186387629782177, −0.36781695494391838827899421765, 2.00428988594600100214551510481, 2.63571475388658294722741457533, 3.93073584593977016905232276533, 4.31850489629127107782029398352, 5.05303213584734202699689688518, 6.10746565822432930091051026768, 6.90046795087838769933835325198, 7.54383883445810561823918469716, 8.383669195273141069950298744852, 9.329269829423901381406771024068

Graph of the ZZ-function along the critical line