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Properties

Label 2-1575-1.1-c3-0-134
Degree 22
Conductor 15751575
Sign −1-1
Analytic cond. 92.928092.9280
Root an. cond. 9.639919.63991
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.53·2-s + 12.5·4-s − 7·7-s + 20.7·8-s + 54.0·11-s − 75.2·13-s − 31.7·14-s − 6.60·16-s − 71.2·17-s − 65.5·19-s + 245.·22-s + 125.·23-s − 341.·26-s − 87.9·28-s − 190.·29-s − 193.·31-s − 195.·32-s − 323.·34-s − 114.·37-s − 297.·38-s − 216.·41-s + 413.·43-s + 679.·44-s + 570.·46-s − 113.·47-s + 49·49-s − 945.·52-s + ⋯
L(s)  = 1  + 1.60·2-s + 1.57·4-s − 0.377·7-s + 0.915·8-s + 1.48·11-s − 1.60·13-s − 0.606·14-s − 0.103·16-s − 1.01·17-s − 0.791·19-s + 2.37·22-s + 1.13·23-s − 2.57·26-s − 0.593·28-s − 1.21·29-s − 1.11·31-s − 1.08·32-s − 1.62·34-s − 0.509·37-s − 1.26·38-s − 0.826·41-s + 1.46·43-s + 2.32·44-s + 1.82·46-s − 0.352·47-s + 0.142·49-s − 2.52·52-s + ⋯

Functional equation

Λ(s)=(1575s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1575s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15751575    =    32⋅52⋅73^{2} \cdot 5^{2} \cdot 7
Sign: −1-1
Analytic conductor: 92.928092.9280
Root analytic conductor: 9.639919.63991
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1575, ( :3/2), −1)(2,\ 1575,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3 1 1
5 1 1
7 1+7T 1 + 7T
good2 1−4.53T+8T2 1 - 4.53T + 8T^{2}
11 1−54.0T+1.33e3T2 1 - 54.0T + 1.33e3T^{2}
13 1+75.2T+2.19e3T2 1 + 75.2T + 2.19e3T^{2}
17 1+71.2T+4.91e3T2 1 + 71.2T + 4.91e3T^{2}
19 1+65.5T+6.85e3T2 1 + 65.5T + 6.85e3T^{2}
23 1−125.T+1.21e4T2 1 - 125.T + 1.21e4T^{2}
29 1+190.T+2.43e4T2 1 + 190.T + 2.43e4T^{2}
31 1+193.T+2.97e4T2 1 + 193.T + 2.97e4T^{2}
37 1+114.T+5.06e4T2 1 + 114.T + 5.06e4T^{2}
41 1+216.T+6.89e4T2 1 + 216.T + 6.89e4T^{2}
43 1−413.T+7.95e4T2 1 - 413.T + 7.95e4T^{2}
47 1+113.T+1.03e5T2 1 + 113.T + 1.03e5T^{2}
53 1−584.T+1.48e5T2 1 - 584.T + 1.48e5T^{2}
59 1+203.T+2.05e5T2 1 + 203.T + 2.05e5T^{2}
61 1+162.T+2.26e5T2 1 + 162.T + 2.26e5T^{2}
67 1+477.T+3.00e5T2 1 + 477.T + 3.00e5T^{2}
71 1+822.T+3.57e5T2 1 + 822.T + 3.57e5T^{2}
73 1−798.T+3.89e5T2 1 - 798.T + 3.89e5T^{2}
79 1+468.T+4.93e5T2 1 + 468.T + 4.93e5T^{2}
83 1+310.T+5.71e5T2 1 + 310.T + 5.71e5T^{2}
89 1+1.31e3T+7.04e5T2 1 + 1.31e3T + 7.04e5T^{2}
97 1−1.31e3T+9.12e5T2 1 - 1.31e3T + 9.12e5T^{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

−8.941664626530402052209743965201, −7.27608393770094668537368206929, −6.92500044824474366830360661971, −6.08445837208276198508666334141, −5.21781657259045096229567898352, −4.39137653105078944536123181915, −3.77033592552118968377450768158, −2.73288405632729858763018071585, −1.83034871472492507952397496032, 0, 1.83034871472492507952397496032, 2.73288405632729858763018071585, 3.77033592552118968377450768158, 4.39137653105078944536123181915, 5.21781657259045096229567898352, 6.08445837208276198508666334141, 6.92500044824474366830360661971, 7.27608393770094668537368206929, 8.941664626530402052209743965201

Graph of the ZZ-function along the critical line