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Properties

Label 2-8550-1.1-c1-0-1
Degree 22
Conductor 85508550
Sign 11
Analytic cond. 68.272068.2720
Root an. cond. 8.262698.26269
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2.69·7-s − 8-s − 5.54·11-s − 2.91·13-s + 2.69·14-s + 16-s − 4.91·17-s + 19-s + 5.54·22-s − 3.60·23-s + 2.91·26-s − 2.69·28-s − 1.08·29-s + 7.54·31-s − 32-s + 4.91·34-s + 4.54·37-s − 38-s − 9.54·43-s − 5.54·44-s + 3.60·46-s + 0.836·47-s + 0.245·49-s − 2.91·52-s − 9.78·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.01·7-s − 0.353·8-s − 1.67·11-s − 0.809·13-s + 0.719·14-s + 0.250·16-s − 1.19·17-s + 0.229·19-s + 1.18·22-s − 0.752·23-s + 0.572·26-s − 0.508·28-s − 0.200·29-s + 1.35·31-s − 0.176·32-s + 0.843·34-s + 0.747·37-s − 0.162·38-s − 1.45·43-s − 0.836·44-s + 0.532·46-s + 0.122·47-s + 0.0350·49-s − 0.404·52-s − 1.34·53-s + ⋯

Functional equation

Λ(s)=(8550s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8550s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 85508550    =    2⋅32⋅52⋅192 \cdot 3^{2} \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 68.272068.2720
Root analytic conductor: 8.262698.26269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8550, ( :1/2), 1)(2,\ 8550,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.21691510820.2169151082
L(12)L(\frac12) ≈\approx 0.21691510820.2169151082
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 1
19 1−T 1 - T
good7 1+2.69T+7T2 1 + 2.69T + 7T^{2}
11 1+5.54T+11T2 1 + 5.54T + 11T^{2}
13 1+2.91T+13T2 1 + 2.91T + 13T^{2}
17 1+4.91T+17T2 1 + 4.91T + 17T^{2}
23 1+3.60T+23T2 1 + 3.60T + 23T^{2}
29 1+1.08T+29T2 1 + 1.08T + 29T^{2}
31 1−7.54T+31T2 1 - 7.54T + 31T^{2}
37 1−4.54T+37T2 1 - 4.54T + 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1+9.54T+43T2 1 + 9.54T + 43T^{2}
47 1−0.836T+47T2 1 - 0.836T + 47T^{2}
53 1+9.78T+53T2 1 + 9.78T + 53T^{2}
59 1+12.9T+59T2 1 + 12.9T + 59T^{2}
61 1+7.38T+61T2 1 + 7.38T + 61T^{2}
67 1−2.85T+67T2 1 - 2.85T + 67T^{2}
71 1+14.4T+71T2 1 + 14.4T + 71T^{2}
73 1−5.15T+73T2 1 - 5.15T + 73T^{2}
79 1+3.09T+79T2 1 + 3.09T + 79T^{2}
83 1+1.71T+83T2 1 + 1.71T + 83T^{2}
89 1−5.09T+89T2 1 - 5.09T + 89T^{2}
97 1+17.2T+97T2 1 + 17.2T + 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

−7.933205542071202177401316833786, −7.16164643299102277471802509098, −6.49546702654402310999984706128, −5.91989154704053771698240690137, −5.00184010162464656513853689742, −4.34745191904989399585305937749, −3.04932136079510884204164235486, −2.73467663422044861728572007819, −1.78326952862109232266535767008, −0.23599053022095932960090973057, 0.23599053022095932960090973057, 1.78326952862109232266535767008, 2.73467663422044861728572007819, 3.04932136079510884204164235486, 4.34745191904989399585305937749, 5.00184010162464656513853689742, 5.91989154704053771698240690137, 6.49546702654402310999984706128, 7.16164643299102277471802509098, 7.933205542071202177401316833786

Graph of the ZZ-function along the critical line