Properties

Label 2-1407-1407.1319-c0-0-0
Degree 22
Conductor 14071407
Sign 0.925+0.378i0.925 + 0.378i
Analytic cond. 0.7021840.702184
Root an. cond. 0.8379640.837964
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.580 + 0.814i)3-s + (0.142 − 0.989i)4-s + (−0.0475 − 0.998i)7-s + (−0.327 + 0.945i)9-s + (0.888 − 0.458i)12-s + (1.16 − 0.600i)13-s + (−0.959 − 0.281i)16-s + (0.143 + 0.124i)19-s + (0.786 − 0.618i)21-s + (−0.888 + 0.458i)25-s + (−0.959 + 0.281i)27-s + (−0.995 − 0.0950i)28-s + (1.61 − 1.03i)31-s + (0.888 + 0.458i)36-s + 0.0951·37-s + ⋯
L(s)  = 1  + (0.580 + 0.814i)3-s + (0.142 − 0.989i)4-s + (−0.0475 − 0.998i)7-s + (−0.327 + 0.945i)9-s + (0.888 − 0.458i)12-s + (1.16 − 0.600i)13-s + (−0.959 − 0.281i)16-s + (0.143 + 0.124i)19-s + (0.786 − 0.618i)21-s + (−0.888 + 0.458i)25-s + (−0.959 + 0.281i)27-s + (−0.995 − 0.0950i)28-s + (1.61 − 1.03i)31-s + (0.888 + 0.458i)36-s + 0.0951·37-s + ⋯

Functional equation

Λ(s)=(1407s/2ΓC(s)L(s)=((0.925+0.378i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1407s/2ΓC(s)L(s)=((0.925+0.378i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14071407    =    37673 \cdot 7 \cdot 67
Sign: 0.925+0.378i0.925 + 0.378i
Analytic conductor: 0.7021840.702184
Root analytic conductor: 0.8379640.837964
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1407(1319,)\chi_{1407} (1319, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1407, ( :0), 0.925+0.378i)(2,\ 1407,\ (\ :0),\ 0.925 + 0.378i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.3656554371.365655437
L(12)L(\frac12) \approx 1.3656554371.365655437
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(0.5800.814i)T 1 + (-0.580 - 0.814i)T
7 1+(0.0475+0.998i)T 1 + (0.0475 + 0.998i)T
67 1+(0.04750.998i)T 1 + (0.0475 - 0.998i)T
good2 1+(0.142+0.989i)T2 1 + (-0.142 + 0.989i)T^{2}
5 1+(0.8880.458i)T2 1 + (0.888 - 0.458i)T^{2}
11 1+(0.841+0.540i)T2 1 + (0.841 + 0.540i)T^{2}
13 1+(1.16+0.600i)T+(0.5800.814i)T2 1 + (-1.16 + 0.600i)T + (0.580 - 0.814i)T^{2}
17 1+(0.2350.971i)T2 1 + (0.235 - 0.971i)T^{2}
19 1+(0.1430.124i)T+(0.142+0.989i)T2 1 + (-0.143 - 0.124i)T + (0.142 + 0.989i)T^{2}
23 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
29 1+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+(1.61+1.03i)T+(0.4150.909i)T2 1 + (-1.61 + 1.03i)T + (0.415 - 0.909i)T^{2}
37 10.0951T+T2 1 - 0.0951T + T^{2}
41 1+(0.235+0.971i)T2 1 + (-0.235 + 0.971i)T^{2}
43 1+(1.07+0.153i)T+(0.9590.281i)T2 1 + (-1.07 + 0.153i)T + (0.959 - 0.281i)T^{2}
47 1+(0.327+0.945i)T2 1 + (-0.327 + 0.945i)T^{2}
53 1+(0.235+0.971i)T2 1 + (0.235 + 0.971i)T^{2}
59 1+(0.995+0.0950i)T2 1 + (-0.995 + 0.0950i)T^{2}
61 1+(1.910.560i)T+(0.8410.540i)T2 1 + (1.91 - 0.560i)T + (0.841 - 0.540i)T^{2}
71 1+(0.723+0.690i)T2 1 + (-0.723 + 0.690i)T^{2}
73 1+(1.341.40i)T+(0.04750.998i)T2 1 + (1.34 - 1.40i)T + (-0.0475 - 0.998i)T^{2}
79 1+(0.9151.77i)T+(0.580+0.814i)T2 1 + (-0.915 - 1.77i)T + (-0.580 + 0.814i)T^{2}
83 1+(0.04750.998i)T2 1 + (0.0475 - 0.998i)T^{2}
89 1+(0.3270.945i)T2 1 + (-0.327 - 0.945i)T^{2}
97 1+(0.580+1.00i)T+(0.50.866i)T2 1 + (-0.580 + 1.00i)T + (-0.5 - 0.866i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.881015897810035803863165780698, −9.069101747719300265267699396625, −8.154262435546445845339174261133, −7.40663482207328874149709720429, −6.24031460469296276692130018724, −5.57078669819718937243750182116, −4.48278792555911361742751908304, −3.81845553794549652178629245485, −2.65940897839852724161540275496, −1.22831713221134968030026347555, 1.72166448917007778309369517735, 2.73564177481551775723597103750, 3.48452945053310921058703218116, 4.57877484337510394353819784972, 6.10405023411206937068844189145, 6.47394956410698699440949465613, 7.58762625485116832485378334498, 8.179104344986139503165865029420, 8.879569972089805198703996439288, 9.340092826008785885035860783214

Graph of the ZZ-function along the critical line