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Properties

Label 2-2550-1.1-c1-0-48
Degree 22
Conductor 25502550
Sign −1-1
Analytic cond. 20.361820.3618
Root an. cond. 4.512414.51241
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 − 3-s + 4-s − 6-s + 2.44·7-s + 8-s + 9-s − 4.89·11-s − 12-s − 6·13-s + 2.44·14-s + 16-s − 17-s + 18-s + 6.89·19-s − 2.44·21-s − 4.89·22-s − 6.44·23-s − 24-s − 6·26-s − 27-s + 2.44·28-s − 9.34·29-s − 6.44·31-s + 32-s + 4.89·33-s − 34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.925·7-s + 0.353·8-s + 0.333·9-s − 1.47·11-s − 0.288·12-s − 1.66·13-s + 0.654·14-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 1.58·19-s − 0.534·21-s − 1.04·22-s − 1.34·23-s − 0.204·24-s − 1.17·26-s − 0.192·27-s + 0.462·28-s − 1.73·29-s − 1.15·31-s + 0.176·32-s + 0.852·33-s − 0.171·34-s + ⋯

Functional equation

Λ(s)=(2550s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2550s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25502550    =    2⋅3⋅52⋅172 \cdot 3 \cdot 5^{2} \cdot 17
Sign: −1-1
Analytic conductor: 20.361820.3618
Root analytic conductor: 4.512414.51241
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2550, ( :1/2), −1)(2,\ 2550,\ (\ :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+T 1 + T
5 1 1
17 1+T 1 + T
good7 1−2.44T+7T2 1 - 2.44T + 7T^{2}
11 1+4.89T+11T2 1 + 4.89T + 11T^{2}
13 1+6T+13T2 1 + 6T + 13T^{2}
19 1−6.89T+19T2 1 - 6.89T + 19T^{2}
23 1+6.44T+23T2 1 + 6.44T + 23T^{2}
29 1+9.34T+29T2 1 + 9.34T + 29T^{2}
31 1+6.44T+31T2 1 + 6.44T + 31T^{2}
37 1−0.449T+37T2 1 - 0.449T + 37T^{2}
41 1+1.10T+41T2 1 + 1.10T + 41T^{2}
43 1−2.89T+43T2 1 - 2.89T + 43T^{2}
47 1+4.89T+47T2 1 + 4.89T + 47T^{2}
53 1−1.10T+53T2 1 - 1.10T + 53T^{2}
59 1−5.79T+59T2 1 - 5.79T + 59T^{2}
61 1−13.3T+61T2 1 - 13.3T + 61T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1−2.44T+71T2 1 - 2.44T + 71T^{2}
73 1+14.8T+73T2 1 + 14.8T + 73T^{2}
79 1+1.55T+79T2 1 + 1.55T + 79T^{2}
83 1+2.89T+83T2 1 + 2.89T + 83T^{2}
89 1+1.79T+89T2 1 + 1.79T + 89T^{2}
97 1−3.79T+97T2 1 - 3.79T + 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

−8.206315397634574447902406415734, −7.45397257341482684004048396476, −7.24214182222067515602782235079, −5.78841642242224113593090383967, −5.30243407290834490264870357348, −4.83414082823285989631889017844, −3.80873989216357019006401529082, −2.60833362751373309240544519149, −1.79684396230397726346931956942, 0, 1.79684396230397726346931956942, 2.60833362751373309240544519149, 3.80873989216357019006401529082, 4.83414082823285989631889017844, 5.30243407290834490264870357348, 5.78841642242224113593090383967, 7.24214182222067515602782235079, 7.45397257341482684004048396476, 8.206315397634574447902406415734

Graph of the ZZ-function along the critical line