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Properties

Label 2-425-5.4-c3-0-28
Degree 22
Conductor 425425
Sign 0.894−0.447i0.894 - 0.447i
Analytic cond. 25.075825.0758
Root an. cond. 5.007575.00757
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.64i·2-s − 5.37i·3-s + 5.31·4-s + 8.81·6-s − 2.20i·7-s + 21.8i·8-s − 1.88·9-s + 18.7·11-s − 28.5i·12-s + 62.9i·13-s + 3.60·14-s + 6.68·16-s + 17i·17-s − 3.09i·18-s + 47.8·19-s + ⋯
L(s)  = 1  + 0.579i·2-s − 1.03i·3-s + 0.663·4-s + 0.599·6-s − 0.118i·7-s + 0.964i·8-s − 0.0698·9-s + 0.514·11-s − 0.686i·12-s + 1.34i·13-s + 0.0689·14-s + 0.104·16-s + 0.242i·17-s − 0.0405i·18-s + 0.577·19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((0.894−0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+3/2) L(s)=((0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: 0.894−0.447i0.894 - 0.447i
Analytic conductor: 25.075825.0758
Root analytic conductor: 5.007575.00757
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ425(324,⋅)\chi_{425} (324, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :3/2), 0.894−0.447i)(2,\ 425,\ (\ :3/2),\ 0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 2.5595474912.559547491
L(12)L(\frac12) ≈\approx 2.5595474912.559547491
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)
bad5 1 1
17 1−17iT 1 - 17iT
good2 1−1.64iT−8T2 1 - 1.64iT - 8T^{2}
3 1+5.37iT−27T2 1 + 5.37iT - 27T^{2}
7 1+2.20iT−343T2 1 + 2.20iT - 343T^{2}
11 1−18.7T+1.33e3T2 1 - 18.7T + 1.33e3T^{2}
13 1−62.9iT−2.19e3T2 1 - 62.9iT - 2.19e3T^{2}
19 1−47.8T+6.85e3T2 1 - 47.8T + 6.85e3T^{2}
23 1−153.iT−1.21e4T2 1 - 153. iT - 1.21e4T^{2}
29 1−64.4T+2.43e4T2 1 - 64.4T + 2.43e4T^{2}
31 1+40.9T+2.97e4T2 1 + 40.9T + 2.97e4T^{2}
37 1+32.6iT−5.06e4T2 1 + 32.6iT - 5.06e4T^{2}
41 1−159.T+6.89e4T2 1 - 159.T + 6.89e4T^{2}
43 1+111.iT−7.95e4T2 1 + 111. iT - 7.95e4T^{2}
47 1+614.iT−1.03e5T2 1 + 614. iT - 1.03e5T^{2}
53 1−308.iT−1.48e5T2 1 - 308. iT - 1.48e5T^{2}
59 1+267.T+2.05e5T2 1 + 267.T + 2.05e5T^{2}
61 1−521.T+2.26e5T2 1 - 521.T + 2.26e5T^{2}
67 1+118.iT−3.00e5T2 1 + 118. iT - 3.00e5T^{2}
71 1−1.14e3T+3.57e5T2 1 - 1.14e3T + 3.57e5T^{2}
73 1+40.7iT−3.89e5T2 1 + 40.7iT - 3.89e5T^{2}
79 1+374.T+4.93e5T2 1 + 374.T + 4.93e5T^{2}
83 1−826.iT−5.71e5T2 1 - 826. iT - 5.71e5T^{2}
89 1−38.9T+7.04e5T2 1 - 38.9T + 7.04e5T^{2}
97 1−917.iT−9.12e5T2 1 - 917. iT - 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

−11.11068090027857833506363755290, −9.831957060178854743437990949131, −8.767976518990893460656588370662, −7.69226416480662668761954316740, −7.05133342759473854331977968521, −6.45728268484996987567925383445, −5.40808572755531706392772493947, −3.87448514002879101247838540678, −2.23294509972993202212436872072, −1.30641850571917803868454402960, 0.956505233259994352199174324282, 2.62800977266097142010115241093, 3.55910511782281963632233152132, 4.64169377381769631497364917854, 5.82553797364197715966600340412, 6.92256857175479013516725102634, 8.014971967377216728859194228964, 9.242437249779718939475982908094, 10.01963762327388303577499394285, 10.64258210614119032539522327220

Graph of the ZZ-function along the critical line