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Properties

Label 2-7448-1.1-c1-0-2
Degree 22
Conductor 74487448
Sign 11
Analytic cond. 59.472559.4725
Root an. cond. 7.711847.71184
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  − 0.786·3-s − 3.29·5-s − 2.38·9-s + 1.29·11-s − 1.21·13-s + 2.59·15-s − 4.08·17-s + 19-s − 8.95·23-s + 5.87·25-s + 4.23·27-s − 9.38·29-s − 1.02·33-s − 2·37-s + 0.954·39-s − 3.57·41-s + 7.72·43-s + 7.85·45-s − 9.46·47-s + 3.21·51-s − 11.9·53-s − 4.27·55-s − 0.786·57-s + 7.21·59-s − 4.87·61-s + 3.99·65-s + 11.3·67-s + ⋯
L(s)  = 1  − 0.454·3-s − 1.47·5-s − 0.793·9-s + 0.391·11-s − 0.336·13-s + 0.669·15-s − 0.990·17-s + 0.229·19-s − 1.86·23-s + 1.17·25-s + 0.814·27-s − 1.74·29-s − 0.177·33-s − 0.328·37-s + 0.152·39-s − 0.558·41-s + 1.17·43-s + 1.17·45-s − 1.38·47-s + 0.449·51-s − 1.64·53-s − 0.576·55-s − 0.104·57-s + 0.939·59-s − 0.623·61-s + 0.496·65-s + 1.39·67-s + ⋯

Functional equation

Λ(s)=(7448s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7448s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 74487448    =    23⋅72⋅192^{3} \cdot 7^{2} \cdot 19
Sign: 11
Analytic conductor: 59.472559.4725
Root analytic conductor: 7.711847.71184
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7448, ( :1/2), 1)(2,\ 7448,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.17065076730.1706507673
L(12)L(\frac12) ≈\approx 0.17065076730.1706507673
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
7 1 1
19 1−T 1 - T
good3 1+0.786T+3T2 1 + 0.786T + 3T^{2}
5 1+3.29T+5T2 1 + 3.29T + 5T^{2}
11 1−1.29T+11T2 1 - 1.29T + 11T^{2}
13 1+1.21T+13T2 1 + 1.21T + 13T^{2}
17 1+4.08T+17T2 1 + 4.08T + 17T^{2}
23 1+8.95T+23T2 1 + 8.95T + 23T^{2}
29 1+9.38T+29T2 1 + 9.38T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1+3.57T+41T2 1 + 3.57T + 41T^{2}
43 1−7.72T+43T2 1 - 7.72T + 43T^{2}
47 1+9.46T+47T2 1 + 9.46T + 47T^{2}
53 1+11.9T+53T2 1 + 11.9T + 53T^{2}
59 1−7.21T+59T2 1 - 7.21T + 59T^{2}
61 1+4.87T+61T2 1 + 4.87T + 61T^{2}
67 1−11.3T+67T2 1 - 11.3T + 67T^{2}
71 1+9.02T+71T2 1 + 9.02T + 71T^{2}
73 1+5.65T+73T2 1 + 5.65T + 73T^{2}
79 1−9.57T+79T2 1 - 9.57T + 79T^{2}
83 1+10.7T+83T2 1 + 10.7T + 83T^{2}
89 1+11.0T+89T2 1 + 11.0T + 89T^{2}
97 1−8.59T+97T2 1 - 8.59T + 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

−7.974531417152484732717604997710, −7.23568322917651081978876098659, −6.54036194293671975182006838754, −5.81778228855105169601843295063, −5.04661974617232001466806677259, −4.19091412997547462066225798548, −3.74876378024051065381116480851, −2.81941899635224422438527585033, −1.74353655445254116995674239693, −0.20408656993888991514899041791, 0.20408656993888991514899041791, 1.74353655445254116995674239693, 2.81941899635224422438527585033, 3.74876378024051065381116480851, 4.19091412997547462066225798548, 5.04661974617232001466806677259, 5.81778228855105169601843295063, 6.54036194293671975182006838754, 7.23568322917651081978876098659, 7.974531417152484732717604997710

Graph of the ZZ-function along the critical line