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Properties

Label 2-26-1.1-c9-0-8
Degree 22
Conductor 2626
Sign −1-1
Analytic cond. 13.390913.3909
Root an. cond. 3.659363.65936
Motivic weight 99
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s + 75·3-s + 256·4-s − 1.97e3·5-s + 1.20e3·6-s − 1.01e4·7-s + 4.09e3·8-s − 1.40e4·9-s − 3.16e4·10-s + 1.88e4·11-s + 1.92e4·12-s + 2.85e4·13-s − 1.61e5·14-s − 1.48e5·15-s + 6.55e4·16-s − 1.42e5·17-s − 2.24e5·18-s + 8.33e4·19-s − 5.06e5·20-s − 7.58e5·21-s + 3.01e5·22-s − 5.36e5·23-s + 3.07e5·24-s + 1.96e6·25-s + 4.56e5·26-s − 2.53e6·27-s − 2.58e6·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.534·3-s + 1/2·4-s − 1.41·5-s + 0.378·6-s − 1.59·7-s + 0.353·8-s − 0.714·9-s − 1.00·10-s + 0.388·11-s + 0.267·12-s + 0.277·13-s − 1.12·14-s − 0.757·15-s + 1/4·16-s − 0.413·17-s − 0.505·18-s + 0.146·19-s − 0.708·20-s − 0.851·21-s + 0.274·22-s − 0.399·23-s + 0.189·24-s + 1.00·25-s + 0.196·26-s − 0.916·27-s − 0.796·28-s + ⋯

Functional equation

Λ(s)=(26s/2 ΓC(s) L(s)=(− Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
Λ(s)=(26s/2 ΓC(s+9/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2626    =    2⋅132 \cdot 13
Sign: −1-1
Analytic conductor: 13.390913.3909
Root analytic conductor: 3.659363.65936
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 26, ( :9/2), −1)(2,\ 26,\ (\ :9/2),\ -1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{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−p4T 1 - p^{4} T
13 1−p4T 1 - p^{4} T
good3 1−25pT+p9T2 1 - 25 p T + p^{9} T^{2}
5 1+1979T+p9T2 1 + 1979 T + p^{9} T^{2}
7 1+1445pT+p9T2 1 + 1445 p T + p^{9} T^{2}
11 1−18850T+p9T2 1 - 18850 T + p^{9} T^{2}
17 1+142403T+p9T2 1 + 142403 T + p^{9} T^{2}
19 1−83302T+p9T2 1 - 83302 T + p^{9} T^{2}
23 1+23328pT+p9T2 1 + 23328 p T + p^{9} T^{2}
29 1+2600442T+p9T2 1 + 2600442 T + p^{9} T^{2}
31 1+2214004T+p9T2 1 + 2214004 T + p^{9} T^{2}
37 1−18099241T+p9T2 1 - 18099241 T + p^{9} T^{2}
41 1−26812240T+p9T2 1 - 26812240 T + p^{9} T^{2}
43 1+42253475T+p9T2 1 + 42253475 T + p^{9} T^{2}
47 1−35914993T+p9T2 1 - 35914993 T + p^{9} T^{2}
53 1+66514064T+p9T2 1 + 66514064 T + p^{9} T^{2}
59 1+108164002T+p9T2 1 + 108164002 T + p^{9} T^{2}
61 1+207449912T+p9T2 1 + 207449912 T + p^{9} T^{2}
67 1−193015514T+p9T2 1 - 193015514 T + p^{9} T^{2}
71 1+201833497T+p9T2 1 + 201833497 T + p^{9} T^{2}
73 1+121628110T+p9T2 1 + 121628110 T + p^{9} T^{2}
79 1−112871912T+p9T2 1 - 112871912 T + p^{9} T^{2}
83 1−308254212T+p9T2 1 - 308254212 T + p^{9} T^{2}
89 1+6374870T+p9T2 1 + 6374870 T + p^{9} T^{2}
97 1−871266886T+p9T2 1 - 871266886 T + p^{9} T^{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

−14.82018513514779883938816757446, −13.44736350363393059801394475741, −12.30808202967032660883118156330, −11.16718089565018451657033949052, −9.253269094196819735286777872271, −7.71431479179701964605801408256, −6.23085352644144134231919182517, −3.97115450154253890308563411468, −3.00825339531731739911485866313, 0, 3.00825339531731739911485866313, 3.97115450154253890308563411468, 6.23085352644144134231919182517, 7.71431479179701964605801408256, 9.253269094196819735286777872271, 11.16718089565018451657033949052, 12.30808202967032660883118156330, 13.44736350363393059801394475741, 14.82018513514779883938816757446

Graph of the ZZ-function along the critical line