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Properties

Label 2-10e3-1.1-c1-0-12
Degree 22
Conductor 10001000
Sign −1-1
Analytic cond. 7.985047.98504
Root an. cond. 2.825782.82578
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.54·3-s − 2.87·7-s + 3.46·9-s + 4.99·11-s − 0.149·13-s + 6.23·17-s − 1.90·19-s + 7.32·21-s − 4.35·23-s − 1.19·27-s − 8.93·29-s − 8.99·31-s − 12.7·33-s + 2.56·37-s + 0.379·39-s + 7.99·41-s − 4.54·43-s − 9.68·47-s + 1.28·49-s − 15.8·51-s − 1.52·53-s + 4.85·57-s + 11.4·59-s − 3.55·61-s − 9.98·63-s − 4.08·67-s + 11.0·69-s + ⋯
L(s)  = 1  − 1.46·3-s − 1.08·7-s + 1.15·9-s + 1.50·11-s − 0.0414·13-s + 1.51·17-s − 0.437·19-s + 1.59·21-s − 0.908·23-s − 0.229·27-s − 1.65·29-s − 1.61·31-s − 2.21·33-s + 0.421·37-s + 0.0608·39-s + 1.24·41-s − 0.692·43-s − 1.41·47-s + 0.184·49-s − 2.21·51-s − 0.209·53-s + 0.642·57-s + 1.49·59-s − 0.455·61-s − 1.25·63-s − 0.498·67-s + 1.33·69-s + ⋯

Functional equation

Λ(s)=(1000s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1000s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10001000    =    23⋅532^{3} \cdot 5^{3}
Sign: −1-1
Analytic conductor: 7.985047.98504
Root analytic conductor: 2.825782.82578
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1000, ( :1/2), −1)(2,\ 1000,\ (\ :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)
bad2 1 1
5 1 1
good3 1+2.54T+3T2 1 + 2.54T + 3T^{2}
7 1+2.87T+7T2 1 + 2.87T + 7T^{2}
11 1−4.99T+11T2 1 - 4.99T + 11T^{2}
13 1+0.149T+13T2 1 + 0.149T + 13T^{2}
17 1−6.23T+17T2 1 - 6.23T + 17T^{2}
19 1+1.90T+19T2 1 + 1.90T + 19T^{2}
23 1+4.35T+23T2 1 + 4.35T + 23T^{2}
29 1+8.93T+29T2 1 + 8.93T + 29T^{2}
31 1+8.99T+31T2 1 + 8.99T + 31T^{2}
37 1−2.56T+37T2 1 - 2.56T + 37T^{2}
41 1−7.99T+41T2 1 - 7.99T + 41T^{2}
43 1+4.54T+43T2 1 + 4.54T + 43T^{2}
47 1+9.68T+47T2 1 + 9.68T + 47T^{2}
53 1+1.52T+53T2 1 + 1.52T + 53T^{2}
59 1−11.4T+59T2 1 - 11.4T + 59T^{2}
61 1+3.55T+61T2 1 + 3.55T + 61T^{2}
67 1+4.08T+67T2 1 + 4.08T + 67T^{2}
71 1+8.56T+71T2 1 + 8.56T + 71T^{2}
73 1+15.2T+73T2 1 + 15.2T + 73T^{2}
79 1+15.0T+79T2 1 + 15.0T + 79T^{2}
83 1−8.96T+83T2 1 - 8.96T + 83T^{2}
89 1+9.26T+89T2 1 + 9.26T + 89T^{2}
97 1−3.66T+97T2 1 - 3.66T + 97T^{2}
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   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

−9.725276904353479422998217414847, −9.006012014627825514812476116523, −7.60548128652200670080916526951, −6.77937199092904670089791612306, −6.01291076837440709499011047570, −5.56643650734157994508055313524, −4.21577703712064417281421256843, −3.40605365645506965289013402652, −1.49123960040188935965338824048, 0, 1.49123960040188935965338824048, 3.40605365645506965289013402652, 4.21577703712064417281421256843, 5.56643650734157994508055313524, 6.01291076837440709499011047570, 6.77937199092904670089791612306, 7.60548128652200670080916526951, 9.006012014627825514812476116523, 9.725276904353479422998217414847

Graph of the ZZ-function along the critical line