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Properties

Label 2-2600-1.1-c1-0-56
Degree 22
Conductor 26002600
Sign −1-1
Analytic cond. 20.761120.7611
Root an. cond. 4.556434.55643
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  + 2.44·3-s − 2·7-s + 2.99·9-s − 4.44·11-s + 13-s − 6.89·17-s + 0.449·19-s − 4.89·21-s − 6.44·23-s + 4·29-s − 4.44·31-s − 10.8·33-s − 4.89·37-s + 2.44·39-s + 10.8·41-s − 11.3·43-s + 2·47-s − 3·49-s − 16.8·51-s − 1.10·53-s + 1.10·57-s + 9.34·59-s − 5.79·61-s − 5.99·63-s + 5.10·67-s − 15.7·69-s − 3.55·71-s + ⋯
L(s)  = 1  + 1.41·3-s − 0.755·7-s + 0.999·9-s − 1.34·11-s + 0.277·13-s − 1.67·17-s + 0.103·19-s − 1.06·21-s − 1.34·23-s + 0.742·29-s − 0.799·31-s − 1.89·33-s − 0.805·37-s + 0.392·39-s + 1.70·41-s − 1.73·43-s + 0.291·47-s − 0.428·49-s − 2.36·51-s − 0.151·53-s + 0.145·57-s + 1.21·59-s − 0.742·61-s − 0.755·63-s + 0.623·67-s − 1.90·69-s − 0.421·71-s + ⋯

Functional equation

Λ(s)=(2600s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2600s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26002600    =    23⋅52⋅132^{3} \cdot 5^{2} \cdot 13
Sign: −1-1
Analytic conductor: 20.761120.7611
Root analytic conductor: 4.556434.55643
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2600, ( :1/2), −1)(2,\ 2600,\ (\ :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 1
5 1 1
13 1−T 1 - T
good3 1−2.44T+3T2 1 - 2.44T + 3T^{2}
7 1+2T+7T2 1 + 2T + 7T^{2}
11 1+4.44T+11T2 1 + 4.44T + 11T^{2}
17 1+6.89T+17T2 1 + 6.89T + 17T^{2}
19 1−0.449T+19T2 1 - 0.449T + 19T^{2}
23 1+6.44T+23T2 1 + 6.44T + 23T^{2}
29 1−4T+29T2 1 - 4T + 29T^{2}
31 1+4.44T+31T2 1 + 4.44T + 31T^{2}
37 1+4.89T+37T2 1 + 4.89T + 37T^{2}
41 1−10.8T+41T2 1 - 10.8T + 41T^{2}
43 1+11.3T+43T2 1 + 11.3T + 43T^{2}
47 1−2T+47T2 1 - 2T + 47T^{2}
53 1+1.10T+53T2 1 + 1.10T + 53T^{2}
59 1−9.34T+59T2 1 - 9.34T + 59T^{2}
61 1+5.79T+61T2 1 + 5.79T + 61T^{2}
67 1−5.10T+67T2 1 - 5.10T + 67T^{2}
71 1+3.55T+71T2 1 + 3.55T + 71T^{2}
73 1−14.6T+73T2 1 - 14.6T + 73T^{2}
79 1−4.89T+79T2 1 - 4.89T + 79T^{2}
83 1+2T+83T2 1 + 2T + 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1+7.79T+97T2 1 + 7.79T + 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

−8.390579709095279782253314366346, −8.016361451205750368657510633918, −7.08969235129423655510865365764, −6.34840803617623298887686004845, −5.32816859233799727954467697358, −4.26634528158627670201356391739, −3.47987180236557990998935249375, −2.62756458243053411118615905809, −2.00783520248972867229596459855, 0, 2.00783520248972867229596459855, 2.62756458243053411118615905809, 3.47987180236557990998935249375, 4.26634528158627670201356391739, 5.32816859233799727954467697358, 6.34840803617623298887686004845, 7.08969235129423655510865365764, 8.016361451205750368657510633918, 8.390579709095279782253314366346

Graph of the ZZ-function along the critical line