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Properties

Label 2-1470-1.1-c3-0-42
Degree 22
Conductor 14701470
Sign −1-1
Analytic cond. 86.732886.7328
Root an. cond. 9.313049.31304
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4·4-s − 5·5-s + 6·6-s − 8·8-s + 9·9-s + 10·10-s − 60·11-s − 12·12-s + 34·13-s + 15·15-s + 16·16-s − 42·17-s − 18·18-s + 76·19-s − 20·20-s + 120·22-s + 24·24-s + 25·25-s − 68·26-s − 27·27-s + 6·29-s − 30·30-s + 232·31-s − 32·32-s + 180·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.64·11-s − 0.288·12-s + 0.725·13-s + 0.258·15-s + 1/4·16-s − 0.599·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 1.16·22-s + 0.204·24-s + 1/5·25-s − 0.512·26-s − 0.192·27-s + 0.0384·29-s − 0.182·30-s + 1.34·31-s − 0.176·32-s + 0.949·33-s + ⋯

Functional equation

Λ(s)=(1470s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1470s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14701470    =    2⋅3⋅5⋅722 \cdot 3 \cdot 5 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 86.732886.7328
Root analytic conductor: 9.313049.31304
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1470, ( :3/2), −1)(2,\ 1470,\ (\ :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)
bad2 1+pT 1 + p T
3 1+pT 1 + p T
5 1+pT 1 + p T
7 1 1
good11 1+60T+p3T2 1 + 60 T + p^{3} T^{2}
13 1−34T+p3T2 1 - 34 T + p^{3} T^{2}
17 1+42T+p3T2 1 + 42 T + p^{3} T^{2}
19 1−4pT+p3T2 1 - 4 p T + p^{3} T^{2}
23 1+p3T2 1 + p^{3} T^{2}
29 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
31 1−232T+p3T2 1 - 232 T + p^{3} T^{2}
37 1−134T+p3T2 1 - 134 T + p^{3} T^{2}
41 1+234T+p3T2 1 + 234 T + p^{3} T^{2}
43 1+412T+p3T2 1 + 412 T + p^{3} T^{2}
47 1−360T+p3T2 1 - 360 T + p^{3} T^{2}
53 1−222T+p3T2 1 - 222 T + p^{3} T^{2}
59 1+660T+p3T2 1 + 660 T + p^{3} T^{2}
61 1−490T+p3T2 1 - 490 T + p^{3} T^{2}
67 1−812T+p3T2 1 - 812 T + p^{3} T^{2}
71 1−120T+p3T2 1 - 120 T + p^{3} T^{2}
73 1+746T+p3T2 1 + 746 T + p^{3} T^{2}
79 1−152T+p3T2 1 - 152 T + p^{3} T^{2}
83 1−804T+p3T2 1 - 804 T + p^{3} T^{2}
89 1−678T+p3T2 1 - 678 T + p^{3} T^{2}
97 1+2pT+p3T2 1 + 2 p T + p^{3} 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

−8.566133886179145134429702390746, −8.020551828485846304147671354679, −7.23660053773631076587518781272, −6.39873574360615936342878832728, −5.46861604953167102930755625526, −4.67623525466600912610991708404, −3.40459617486998974543510152713, −2.38550360697533715404005323816, −1.00941817969652548790483069606, 0, 1.00941817969652548790483069606, 2.38550360697533715404005323816, 3.40459617486998974543510152713, 4.67623525466600912610991708404, 5.46861604953167102930755625526, 6.39873574360615936342878832728, 7.23660053773631076587518781272, 8.020551828485846304147671354679, 8.566133886179145134429702390746

Graph of the ZZ-function along the critical line