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Properties

Label 2-1700-1700.1271-c0-0-1
Degree 22
Conductor 17001700
Sign 0.862−0.506i0.862 - 0.506i
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  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (0.809 − 0.587i)5-s + (0.587 + 0.809i)8-s + (−0.951 + 0.309i)9-s + (0.951 − 0.309i)10-s + (0.363 + 1.11i)13-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s − 0.999·18-s + 0.999·20-s + (0.309 − 0.951i)25-s + 1.17i·26-s + (0.142 + 0.896i)29-s + i·32-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (0.809 − 0.587i)5-s + (0.587 + 0.809i)8-s + (−0.951 + 0.309i)9-s + (0.951 − 0.309i)10-s + (0.363 + 1.11i)13-s + (0.309 + 0.951i)16-s + (−0.309 − 0.951i)17-s − 0.999·18-s + 0.999·20-s + (0.309 − 0.951i)25-s + 1.17i·26-s + (0.142 + 0.896i)29-s + i·32-s + ⋯

Functional equation

Λ(s)=(1700s/2 ΓC(s) L(s)=((0.862−0.506i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1700s/2 ΓC(s) L(s)=((0.862−0.506i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17001700    =    22⋅52⋅172^{2} \cdot 5^{2} \cdot 17
Sign: 0.862−0.506i0.862 - 0.506i
Analytic conductor: 0.8484100.848410
Root analytic conductor: 0.9210920.921092
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1700(1271,⋅)\chi_{1700} (1271, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1700, ( :0), 0.862−0.506i)(2,\ 1700,\ (\ :0),\ 0.862 - 0.506i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.1786084182.178608418
L(12)L(\frac12) ≈\approx 2.1786084182.178608418
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+(−0.951−0.309i)T 1 + (-0.951 - 0.309i)T
5 1+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
17 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
good3 1+(0.951−0.309i)T2 1 + (0.951 - 0.309i)T^{2}
7 1+iT2 1 + iT^{2}
11 1+(0.587−0.809i)T2 1 + (0.587 - 0.809i)T^{2}
13 1+(−0.363−1.11i)T+(−0.809+0.587i)T2 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2}
19 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
23 1+(0.587−0.809i)T2 1 + (0.587 - 0.809i)T^{2}
29 1+(−0.142−0.896i)T+(−0.951+0.309i)T2 1 + (-0.142 - 0.896i)T + (-0.951 + 0.309i)T^{2}
31 1+(−0.951−0.309i)T2 1 + (-0.951 - 0.309i)T^{2}
37 1+(−0.809+1.58i)T+(−0.587−0.809i)T2 1 + (-0.809 + 1.58i)T + (-0.587 - 0.809i)T^{2}
41 1+(1.76+0.896i)T+(0.587+0.809i)T2 1 + (1.76 + 0.896i)T + (0.587 + 0.809i)T^{2}
43 1+T2 1 + T^{2}
47 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
53 1+(0.690−0.951i)T+(−0.309−0.951i)T2 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2}
59 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
61 1+(0.896+1.76i)T+(−0.587+0.809i)T2 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2}
67 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
71 1+(0.951−0.309i)T2 1 + (0.951 - 0.309i)T^{2}
73 1+(0.278−0.142i)T+(0.587−0.809i)T2 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2}
79 1+(−0.951+0.309i)T2 1 + (-0.951 + 0.309i)T^{2}
83 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
89 1+(0.587−1.80i)T+(−0.809−0.587i)T2 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2}
97 1+(−0.278−1.76i)T+(−0.951+0.309i)T2 1 + (-0.278 - 1.76i)T + (-0.951 + 0.309i)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

−9.343007597449553080060098437759, −8.839573348371055477353546059787, −7.975160184183540512703303826268, −6.92146810719295920132731601460, −6.29311748627822302911422693490, −5.37655631530592154541727317910, −4.90386069319595101373230451012, −3.83979389728859768296722342993, −2.68568662589222422060577577307, −1.81012166213159151054660497339, 1.54229158408636676130282228954, 2.80214802724370521515956217042, 3.26181198649213658708302791762, 4.49984570788973186341281083380, 5.55600479057937034571006726346, 6.08708668275975580834125233031, 6.64253660407338528837044053857, 7.83368530234974541875017519008, 8.650708821574018866282941529601, 9.843471179915113507844852175649

Graph of the ZZ-function along the critical line