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Properties

Label 2-1700-1700.1239-c0-0-0
Degree 22
Conductor 17001700
Sign 0.837+0.546i0.837 + 0.546i
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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.156 + 0.987i)2-s + (−0.951 − 0.309i)4-s + (−0.951 + 0.309i)5-s + (0.453 − 0.891i)8-s + (−0.987 + 0.156i)9-s + (−0.156 − 0.987i)10-s + (−0.734 − 0.533i)13-s + (0.809 + 0.587i)16-s + (0.587 − 0.809i)17-s − i·18-s + 20-s + (0.809 − 0.587i)25-s + (0.642 − 0.642i)26-s + (−0.152 − 1.93i)29-s + (−0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s + (−0.951 − 0.309i)4-s + (−0.951 + 0.309i)5-s + (0.453 − 0.891i)8-s + (−0.987 + 0.156i)9-s + (−0.156 − 0.987i)10-s + (−0.734 − 0.533i)13-s + (0.809 + 0.587i)16-s + (0.587 − 0.809i)17-s − i·18-s + 20-s + (0.809 − 0.587i)25-s + (0.642 − 0.642i)26-s + (−0.152 − 1.93i)29-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

Λ(s)=(1700s/2 ΓC(s) L(s)=((0.837+0.546i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1700s/2 ΓC(s) L(s)=((0.837+0.546i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17001700    =    22⋅52⋅172^{2} \cdot 5^{2} \cdot 17
Sign: 0.837+0.546i0.837 + 0.546i
Analytic conductor: 0.8484100.848410
Root analytic conductor: 0.9210920.921092
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1700(1239,⋅)\chi_{1700} (1239, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1700, ( :0), 0.837+0.546i)(2,\ 1700,\ (\ :0),\ 0.837 + 0.546i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.44751361180.4475136118
L(12)L(\frac12) ≈\approx 0.44751361180.4475136118
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+(0.156−0.987i)T 1 + (0.156 - 0.987i)T
5 1+(0.951−0.309i)T 1 + (0.951 - 0.309i)T
17 1+(−0.587+0.809i)T 1 + (-0.587 + 0.809i)T
good3 1+(0.987−0.156i)T2 1 + (0.987 - 0.156i)T^{2}
7 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
11 1+(0.891−0.453i)T2 1 + (0.891 - 0.453i)T^{2}
13 1+(0.734+0.533i)T+(0.309+0.951i)T2 1 + (0.734 + 0.533i)T + (0.309 + 0.951i)T^{2}
19 1+(0.587+0.809i)T2 1 + (0.587 + 0.809i)T^{2}
23 1+(−0.891+0.453i)T2 1 + (-0.891 + 0.453i)T^{2}
29 1+(0.152+1.93i)T+(−0.987+0.156i)T2 1 + (0.152 + 1.93i)T + (-0.987 + 0.156i)T^{2}
31 1+(−0.156+0.987i)T2 1 + (-0.156 + 0.987i)T^{2}
37 1+(−1.65−0.398i)T+(0.891+0.453i)T2 1 + (-1.65 - 0.398i)T + (0.891 + 0.453i)T^{2}
41 1+(0.678+1.10i)T+(−0.453+0.891i)T2 1 + (0.678 + 1.10i)T + (-0.453 + 0.891i)T^{2}
43 1−iT2 1 - iT^{2}
47 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
53 1+(0.809+1.58i)T+(−0.587+0.809i)T2 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2}
59 1+(−0.951+0.309i)T2 1 + (-0.951 + 0.309i)T^{2}
61 1+(1.47−0.355i)T+(0.891−0.453i)T2 1 + (1.47 - 0.355i)T + (0.891 - 0.453i)T^{2}
67 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
71 1+(−0.987+0.156i)T2 1 + (-0.987 + 0.156i)T^{2}
73 1+(−1.70−1.04i)T+(0.453+0.891i)T2 1 + (-1.70 - 1.04i)T + (0.453 + 0.891i)T^{2}
79 1+(−0.156−0.987i)T2 1 + (-0.156 - 0.987i)T^{2}
83 1+(−0.587−0.809i)T2 1 + (-0.587 - 0.809i)T^{2}
89 1+(−0.183−0.253i)T+(−0.309+0.951i)T2 1 + (-0.183 - 0.253i)T + (-0.309 + 0.951i)T^{2}
97 1+(−1.04+0.0819i)T+(0.987−0.156i)T2 1 + (-1.04 + 0.0819i)T + (0.987 - 0.156i)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

−9.402190789253013735996695668107, −8.209222315077690750705044482081, −8.010096287574154643724402043946, −7.19615383789917107806842717841, −6.33011986735033216411654473308, −5.44107354805166303661383394022, −4.70126971792462970312153583549, −3.66773852836896879621540260379, −2.66301666039660477451592423566, −0.38918764602838744874940965603, 1.36702451386486016297676511654, 2.80571160727100382348653681224, 3.51644639758850623918155467888, 4.50283765212201168111905084458, 5.20532648779213072476315172581, 6.35267410311366254381686768548, 7.65472362103919808691679564798, 8.076665530767715228733437189611, 9.031992587070554457544887103388, 9.427869986604661473746570776095

Graph of the ZZ-function along the critical line