↑

Properties

Label 2-3276-91.34-c1-0-9
Degree 22
Conductor 32763276
Sign −0.518−0.855i-0.518 - 0.855i
Analytic cond. 26.158926.1589
Root an. cond. 5.114585.11458
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  + (1.64 − 1.64i)5-s + (−2.60 + 0.468i)7-s + (2.25 − 2.25i)11-s + (−2.84 + 2.21i)13-s + 0.368·17-s + (−3.97 + 3.97i)19-s − 1.42i·23-s − 0.421i·25-s − 4.07·29-s + (−1.27 + 1.27i)31-s + (−3.51 + 5.05i)35-s + (−2.46 + 2.46i)37-s + (−6.19 + 6.19i)41-s − 2.85i·43-s + (−1.71 − 1.71i)47-s + ⋯
L(s)  = 1  + (0.736 − 0.736i)5-s + (−0.984 + 0.177i)7-s + (0.680 − 0.680i)11-s + (−0.789 + 0.613i)13-s + 0.0893·17-s + (−0.912 + 0.912i)19-s − 0.296i·23-s − 0.0843i·25-s − 0.756·29-s + (−0.229 + 0.229i)31-s + (−0.594 + 0.855i)35-s + (−0.405 + 0.405i)37-s + (−0.966 + 0.966i)41-s − 0.436i·43-s + (−0.250 − 0.250i)47-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=((−0.518−0.855i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=((−0.518−0.855i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: −0.518−0.855i-0.518 - 0.855i
Analytic conductor: 26.158926.1589
Root analytic conductor: 5.114585.11458
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3276(1945,⋅)\chi_{3276} (1945, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3276, ( :1/2), −0.518−0.855i)(2,\ 3276,\ (\ :1/2),\ -0.518 - 0.855i)

Particular Values

L(1)L(1) ≈\approx 0.60456181990.6045618199
L(12)L(\frac12) ≈\approx 0.60456181990.6045618199
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
7 1+(2.60−0.468i)T 1 + (2.60 - 0.468i)T
13 1+(2.84−2.21i)T 1 + (2.84 - 2.21i)T
good5 1+(−1.64+1.64i)T−5iT2 1 + (-1.64 + 1.64i)T - 5iT^{2}
11 1+(−2.25+2.25i)T−11iT2 1 + (-2.25 + 2.25i)T - 11iT^{2}
17 1−0.368T+17T2 1 - 0.368T + 17T^{2}
19 1+(3.97−3.97i)T−19iT2 1 + (3.97 - 3.97i)T - 19iT^{2}
23 1+1.42iT−23T2 1 + 1.42iT - 23T^{2}
29 1+4.07T+29T2 1 + 4.07T + 29T^{2}
31 1+(1.27−1.27i)T−31iT2 1 + (1.27 - 1.27i)T - 31iT^{2}
37 1+(2.46−2.46i)T−37iT2 1 + (2.46 - 2.46i)T - 37iT^{2}
41 1+(6.19−6.19i)T−41iT2 1 + (6.19 - 6.19i)T - 41iT^{2}
43 1+2.85iT−43T2 1 + 2.85iT - 43T^{2}
47 1+(1.71+1.71i)T+47iT2 1 + (1.71 + 1.71i)T + 47iT^{2}
53 1−5.15T+53T2 1 - 5.15T + 53T^{2}
59 1+(0.539+0.539i)T+59iT2 1 + (0.539 + 0.539i)T + 59iT^{2}
61 1−8.98iT−61T2 1 - 8.98iT - 61T^{2}
67 1+(5.51+5.51i)T+67iT2 1 + (5.51 + 5.51i)T + 67iT^{2}
71 1+(−6.96−6.96i)T+71iT2 1 + (-6.96 - 6.96i)T + 71iT^{2}
73 1+(−3.21−3.21i)T+73iT2 1 + (-3.21 - 3.21i)T + 73iT^{2}
79 1+3.28T+79T2 1 + 3.28T + 79T^{2}
83 1+(12.1−12.1i)T−83iT2 1 + (12.1 - 12.1i)T - 83iT^{2}
89 1+(6.35+6.35i)T+89iT2 1 + (6.35 + 6.35i)T + 89iT^{2}
97 1+(−3.00+3.00i)T−97iT2 1 + (-3.00 + 3.00i)T - 97iT^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−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

−8.809117169639859994728435471162, −8.503644223424458798000779879142, −7.25322628025316920617260902607, −6.54879536357701514934449265830, −5.89927177043033493503594977233, −5.22239853477948136729742404905, −4.20346899378554503326790724231, −3.41256220261119889901651912788, −2.29236111145540152207166822811, −1.34034128461277982605989460540, 0.17402810807337805389116016637, 1.88959479774717505928039692866, 2.67779117988115777220894619554, 3.54199403361136348400901764358, 4.48333763260048803039153956274, 5.49491971659226126620168401340, 6.23717558610037697929125270436, 6.95022577926228565574974265842, 7.30630436342463778477432584569, 8.488155150230438463014815230766

Graph of the ZZ-function along the critical line