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Properties

Label 2-1700-1700.1467-c0-0-0
Degree 22
Conductor 17001700
Sign −0.0231+0.999i-0.0231 + 0.999i
Analytic cond. 0.8484100.848410
Root an. cond. 0.9210920.921092
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.996 − 0.0784i)2-s + (0.987 + 0.156i)4-s + (0.233 − 0.972i)5-s + (−0.972 − 0.233i)8-s + (0.649 − 0.760i)9-s + (−0.309 + 0.951i)10-s + (0.322 − 0.993i)13-s + (0.951 + 0.309i)16-s + (−0.987 − 0.156i)17-s + (−0.707 + 0.707i)18-s + (0.382 − 0.923i)20-s + (−0.891 − 0.453i)25-s + (−0.399 + 0.965i)26-s + (0.0984 + 0.213i)29-s + (−0.923 − 0.382i)32-s + ⋯
L(s)  = 1  + (−0.996 − 0.0784i)2-s + (0.987 + 0.156i)4-s + (0.233 − 0.972i)5-s + (−0.972 − 0.233i)8-s + (0.649 − 0.760i)9-s + (−0.309 + 0.951i)10-s + (0.322 − 0.993i)13-s + (0.951 + 0.309i)16-s + (−0.987 − 0.156i)17-s + (−0.707 + 0.707i)18-s + (0.382 − 0.923i)20-s + (−0.891 − 0.453i)25-s + (−0.399 + 0.965i)26-s + (0.0984 + 0.213i)29-s + (−0.923 − 0.382i)32-s + ⋯

Functional equation

Λ(s)=(1700s/2 ΓC(s) L(s)=((−0.0231+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0231 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1700s/2 ΓC(s) L(s)=((−0.0231+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0231 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17001700    =    22⋅52⋅172^{2} \cdot 5^{2} \cdot 17
Sign: −0.0231+0.999i-0.0231 + 0.999i
Analytic conductor: 0.8484100.848410
Root analytic conductor: 0.9210920.921092
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1700(1467,⋅)\chi_{1700} (1467, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1700, ( :0), −0.0231+0.999i)(2,\ 1700,\ (\ :0),\ -0.0231 + 0.999i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.73463161300.7346316130
L(12)L(\frac12) ≈\approx 0.73463161300.7346316130
L(1)L(1) 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+(0.996+0.0784i)T 1 + (0.996 + 0.0784i)T
5 1+(−0.233+0.972i)T 1 + (-0.233 + 0.972i)T
17 1+(0.987+0.156i)T 1 + (0.987 + 0.156i)T
good3 1+(−0.649+0.760i)T2 1 + (-0.649 + 0.760i)T^{2}
7 1+(0.382+0.923i)T2 1 + (0.382 + 0.923i)T^{2}
11 1+(0.852+0.522i)T2 1 + (0.852 + 0.522i)T^{2}
13 1+(−0.322+0.993i)T+(−0.809−0.587i)T2 1 + (-0.322 + 0.993i)T + (-0.809 - 0.587i)T^{2}
19 1+(−0.891−0.453i)T2 1 + (-0.891 - 0.453i)T^{2}
23 1+(−0.522+0.852i)T2 1 + (-0.522 + 0.852i)T^{2}
29 1+(−0.0984−0.213i)T+(−0.649+0.760i)T2 1 + (-0.0984 - 0.213i)T + (-0.649 + 0.760i)T^{2}
31 1+(−0.996−0.0784i)T2 1 + (-0.996 - 0.0784i)T^{2}
37 1+(−0.473−0.265i)T+(0.522+0.852i)T2 1 + (-0.473 - 0.265i)T + (0.522 + 0.852i)T^{2}
41 1+(0.831+0.0984i)T+(0.972+0.233i)T2 1 + (0.831 + 0.0984i)T + (0.972 + 0.233i)T^{2}
43 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
47 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
53 1+(0.453−0.108i)T+(0.891−0.453i)T2 1 + (0.453 - 0.108i)T + (0.891 - 0.453i)T^{2}
59 1+(−0.156−0.987i)T2 1 + (-0.156 - 0.987i)T^{2}
61 1+(−0.916−1.63i)T+(−0.522+0.852i)T2 1 + (-0.916 - 1.63i)T + (-0.522 + 0.852i)T^{2}
67 1+(−0.951+0.309i)T2 1 + (-0.951 + 0.309i)T^{2}
71 1+(−0.760−0.649i)T2 1 + (-0.760 - 0.649i)T^{2}
73 1+(−0.0486+0.0616i)T+(−0.233−0.972i)T2 1 + (-0.0486 + 0.0616i)T + (-0.233 - 0.972i)T^{2}
79 1+(0.996−0.0784i)T2 1 + (0.996 - 0.0784i)T^{2}
83 1+(−0.453+0.891i)T2 1 + (-0.453 + 0.891i)T^{2}
89 1+(−0.690+1.35i)T+(−0.587−0.809i)T2 1 + (-0.690 + 1.35i)T + (-0.587 - 0.809i)T^{2}
97 1+(−1.63−0.603i)T+(0.760+0.649i)T2 1 + (-1.63 - 0.603i)T + (0.760 + 0.649i)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

−9.256910990375064246238947225754, −8.653251332635037685192634490648, −7.992854057391472718331419695804, −7.03807783951414100123849514344, −6.28493641790589486322975363989, −5.39916530846693251942847457135, −4.27837085611380528909269129353, −3.19228968529958719625287581219, −1.89601992062369554032376323186, −0.790077141867558966896586057069, 1.72991124010168544348519011316, 2.43412002997145877748232174260, 3.67764352335201610649192486485, 4.85175908585683672975056477803, 6.14329901629636612452447311143, 6.66805970225962176343153532106, 7.37439763746551409002586795325, 8.119442327443998737559756807442, 9.029132275987728047619814637390, 9.751590460515881259123362486492

Graph of the ZZ-function along the critical line