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Properties

Label 2-9702-1.1-c1-0-105
Degree 22
Conductor 97029702
Sign −1-1
Analytic cond. 77.470877.4708
Root an. cond. 8.801758.80175
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 − 2·5-s − 8-s + 2·10-s + 11-s + 2.58·13-s + 16-s − 2·17-s + 6.24·19-s − 2·20-s − 22-s − 0.828·23-s − 25-s − 2.58·26-s + 1.65·29-s − 2.24·31-s − 32-s + 2·34-s − 4.82·37-s − 6.24·38-s + 2·40-s − 0.343·41-s + 0.828·43-s + 44-s + 0.828·46-s − 11.8·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 0.301·11-s + 0.717·13-s + 0.250·16-s − 0.485·17-s + 1.43·19-s − 0.447·20-s − 0.213·22-s − 0.172·23-s − 0.200·25-s − 0.507·26-s + 0.307·29-s − 0.402·31-s − 0.176·32-s + 0.342·34-s − 0.793·37-s − 1.01·38-s + 0.316·40-s − 0.0535·41-s + 0.126·43-s + 0.150·44-s + 0.122·46-s − 1.73·47-s + ⋯

Functional equation

Λ(s)=(9702s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9702s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 97029702    =    2⋅32⋅72⋅112 \cdot 3^{2} \cdot 7^{2} \cdot 11
Sign: −1-1
Analytic conductor: 77.470877.4708
Root analytic conductor: 8.801758.80175
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9702, ( :1/2), −1)(2,\ 9702,\ (\ :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
7 1 1
11 1−T 1 - T
good5 1+2T+5T2 1 + 2T + 5T^{2}
13 1−2.58T+13T2 1 - 2.58T + 13T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1−6.24T+19T2 1 - 6.24T + 19T^{2}
23 1+0.828T+23T2 1 + 0.828T + 23T^{2}
29 1−1.65T+29T2 1 - 1.65T + 29T^{2}
31 1+2.24T+31T2 1 + 2.24T + 31T^{2}
37 1+4.82T+37T2 1 + 4.82T + 37T^{2}
41 1+0.343T+41T2 1 + 0.343T + 41T^{2}
43 1−0.828T+43T2 1 - 0.828T + 43T^{2}
47 1+11.8T+47T2 1 + 11.8T + 47T^{2}
53 1+6.48T+53T2 1 + 6.48T + 53T^{2}
59 1+1.17T+59T2 1 + 1.17T + 59T^{2}
61 1−5.41T+61T2 1 - 5.41T + 61T^{2}
67 1+6.82T+67T2 1 + 6.82T + 67T^{2}
71 1−5.65T+71T2 1 - 5.65T + 71T^{2}
73 1−0.343T+73T2 1 - 0.343T + 73T^{2}
79 1−0.485T+79T2 1 - 0.485T + 79T^{2}
83 1−9.07T+83T2 1 - 9.07T + 83T^{2}
89 1+12.2T+89T2 1 + 12.2T + 89T^{2}
97 1−9.89T+97T2 1 - 9.89T + 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

−7.46097777734601280385088201311, −6.82281213986347351775868191746, −6.16149469322688461261825954011, −5.32633217586220673048508144680, −4.50578519622270047206322049979, −3.60037246306004207729214130025, −3.17455506569084298960814445469, −1.96617408269351178127910299745, −1.08946590599935230753533481686, 0, 1.08946590599935230753533481686, 1.96617408269351178127910299745, 3.17455506569084298960814445469, 3.60037246306004207729214130025, 4.50578519622270047206322049979, 5.32633217586220673048508144680, 6.16149469322688461261825954011, 6.82281213986347351775868191746, 7.46097777734601280385088201311

Graph of the ZZ-function along the critical line