↑

Properties

Label 2-845-1.1-c1-0-25
Degree 22
Conductor 845845
Sign −1-1
Analytic cond. 6.747356.74735
Root an. cond. 2.597562.59756
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.445·2-s − 3.24·3-s − 1.80·4-s − 5-s − 1.44·6-s + 3.24·7-s − 1.69·8-s + 7.54·9-s − 0.445·10-s − 0.692·11-s + 5.85·12-s + 1.44·14-s + 3.24·15-s + 2.85·16-s + 3.74·17-s + 3.35·18-s − 1.53·19-s + 1.80·20-s − 10.5·21-s − 0.307·22-s − 1.22·23-s + 5.49·24-s + 25-s − 14.7·27-s − 5.85·28-s − 6.07·29-s + 1.44·30-s + ⋯
L(s)  = 1  + 0.314·2-s − 1.87·3-s − 0.900·4-s − 0.447·5-s − 0.589·6-s + 1.22·7-s − 0.598·8-s + 2.51·9-s − 0.140·10-s − 0.208·11-s + 1.68·12-s + 0.386·14-s + 0.838·15-s + 0.712·16-s + 0.907·17-s + 0.791·18-s − 0.351·19-s + 0.402·20-s − 2.30·21-s − 0.0656·22-s − 0.255·23-s + 1.12·24-s + 0.200·25-s − 2.83·27-s − 1.10·28-s − 1.12·29-s + 0.263·30-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 845, ( :1/2), −1)(2,\ 845,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad5 1+T 1 + T
13 1 1
good2 1−0.445T+2T2 1 - 0.445T + 2T^{2}
3 1+3.24T+3T2 1 + 3.24T + 3T^{2}
7 1−3.24T+7T2 1 - 3.24T + 7T^{2}
11 1+0.692T+11T2 1 + 0.692T + 11T^{2}
17 1−3.74T+17T2 1 - 3.74T + 17T^{2}
19 1+1.53T+19T2 1 + 1.53T + 19T^{2}
23 1+1.22T+23T2 1 + 1.22T + 23T^{2}
29 1+6.07T+29T2 1 + 6.07T + 29T^{2}
31 1+8.45T+31T2 1 + 8.45T + 31T^{2}
37 1+1.89T+37T2 1 + 1.89T + 37T^{2}
41 1+0.457T+41T2 1 + 0.457T + 41T^{2}
43 1+6.19T+43T2 1 + 6.19T + 43T^{2}
47 1−11.5T+47T2 1 - 11.5T + 47T^{2}
53 1−0.801T+53T2 1 - 0.801T + 53T^{2}
59 1−6.60T+59T2 1 - 6.60T + 59T^{2}
61 1+4.19T+61T2 1 + 4.19T + 61T^{2}
67 1+13.8T+67T2 1 + 13.8T + 67T^{2}
71 1+9.87T+71T2 1 + 9.87T + 71T^{2}
73 1−8.05T+73T2 1 - 8.05T + 73T^{2}
79 1+16.5T+79T2 1 + 16.5T + 79T^{2}
83 1−6.17T+83T2 1 - 6.17T + 83T^{2}
89 1+10.5T+89T2 1 + 10.5T + 89T^{2}
97 1+3.45T+97T2 1 + 3.45T + 97T^{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

−10.08260074819754363975525712483, −8.980806120475066550950734163951, −7.87742937547111830405240015715, −7.15987609639137326358242594432, −5.78516715931268228571650472370, −5.38732873807321509278610472570, −4.57390845294357769008725299808, −3.82014254941757263839934566899, −1.43288330913151538286107058378, 0, 1.43288330913151538286107058378, 3.82014254941757263839934566899, 4.57390845294357769008725299808, 5.38732873807321509278610472570, 5.78516715931268228571650472370, 7.15987609639137326358242594432, 7.87742937547111830405240015715, 8.980806120475066550950734163951, 10.08260074819754363975525712483

Graph of the ZZ-function along the critical line