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Properties

Label 2-1710-1.1-c1-0-25
Degree 22
Conductor 17101710
Sign −1-1
Analytic cond. 13.654413.6544
Root an. cond. 3.695183.69518
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-s + 4-s − 5-s − 2.56·7-s + 8-s − 10-s − 4·11-s + 5.68·13-s − 2.56·14-s + 16-s − 3.43·17-s − 19-s − 20-s − 4·22-s − 7.68·23-s + 25-s + 5.68·26-s − 2.56·28-s + 5.68·29-s − 5.12·31-s + 32-s − 3.43·34-s + 2.56·35-s − 6·37-s − 38-s − 40-s − 12.2·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.968·7-s + 0.353·8-s − 0.316·10-s − 1.20·11-s + 1.57·13-s − 0.684·14-s + 0.250·16-s − 0.833·17-s − 0.229·19-s − 0.223·20-s − 0.852·22-s − 1.60·23-s + 0.200·25-s + 1.11·26-s − 0.484·28-s + 1.05·29-s − 0.920·31-s + 0.176·32-s − 0.589·34-s + 0.432·35-s − 0.986·37-s − 0.162·38-s − 0.158·40-s − 1.91·41-s + ⋯

Functional equation

Λ(s)=(1710s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1710s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17101710    =    2⋅32⋅5⋅192 \cdot 3^{2} \cdot 5 \cdot 19
Sign: −1-1
Analytic conductor: 13.654413.6544
Root analytic conductor: 3.695183.69518
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1710, ( :1/2), −1)(2,\ 1710,\ (\ :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−T 1 - T
3 1 1
5 1+T 1 + T
19 1+T 1 + T
good7 1+2.56T+7T2 1 + 2.56T + 7T^{2}
11 1+4T+11T2 1 + 4T + 11T^{2}
13 1−5.68T+13T2 1 - 5.68T + 13T^{2}
17 1+3.43T+17T2 1 + 3.43T + 17T^{2}
23 1+7.68T+23T2 1 + 7.68T + 23T^{2}
29 1−5.68T+29T2 1 - 5.68T + 29T^{2}
31 1+5.12T+31T2 1 + 5.12T + 31T^{2}
37 1+6T+37T2 1 + 6T + 37T^{2}
41 1+12.2T+41T2 1 + 12.2T + 41T^{2}
43 1+2.87T+43T2 1 + 2.87T + 43T^{2}
47 1+6.24T+47T2 1 + 6.24T + 47T^{2}
53 1−4.56T+53T2 1 - 4.56T + 53T^{2}
59 1+2.56T+59T2 1 + 2.56T + 59T^{2}
61 1−11.1T+61T2 1 - 11.1T + 61T^{2}
67 1+2.56T+67T2 1 + 2.56T + 67T^{2}
71 1+10.2T+71T2 1 + 10.2T + 71T^{2}
73 1+1.68T+73T2 1 + 1.68T + 73T^{2}
79 1+5.12T+79T2 1 + 5.12T + 79T^{2}
83 1+2.87T+83T2 1 + 2.87T + 83T^{2}
89 1+2T+89T2 1 + 2T + 89T^{2}
97 1−6T+97T2 1 - 6T + 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

−8.659425829008491346626045798090, −8.260244476163765653547203711503, −7.13661433699294168140337335468, −6.42321328901943745628048326543, −5.74582432695378909368203368337, −4.74006071669223067052104963608, −3.74895412216667020541456557883, −3.15229793390407648459611943934, −1.92056950891894982430089685637, 0, 1.92056950891894982430089685637, 3.15229793390407648459611943934, 3.74895412216667020541456557883, 4.74006071669223067052104963608, 5.74582432695378909368203368337, 6.42321328901943745628048326543, 7.13661433699294168140337335468, 8.260244476163765653547203711503, 8.659425829008491346626045798090

Graph of the ZZ-function along the critical line