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Properties

Label 2-1700-340.339-c0-0-3
Degree 22
Conductor 17001700
Sign 0.894+0.447i0.894 + 0.447i
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  − i·2-s − 4-s + i·8-s + 9-s + 2i·13-s + 16-s + i·17-s − i·18-s + 2·26-s − i·32-s + 34-s − 36-s + 49-s − 2i·52-s − 2i·53-s + ⋯
L(s)  = 1  − i·2-s − 4-s + i·8-s + 9-s + 2i·13-s + 16-s + i·17-s − i·18-s + 2·26-s − i·32-s + 34-s − 36-s + 49-s − 2i·52-s − 2i·53-s + ⋯

Functional equation

Λ(s)=(1700s/2 ΓC(s) L(s)=((0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1700s/2 ΓC(s) L(s)=((0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17001700    =    22⋅52⋅172^{2} \cdot 5^{2} \cdot 17
Sign: 0.894+0.447i0.894 + 0.447i
Analytic conductor: 0.8484100.848410
Root analytic conductor: 0.9210920.921092
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1700(1699,⋅)\chi_{1700} (1699, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1700, ( :0), 0.894+0.447i)(2,\ 1700,\ (\ :0),\ 0.894 + 0.447i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0588293391.058829339
L(12)L(\frac12) ≈\approx 1.0588293391.058829339
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+iT 1 + iT
5 1 1
17 1−iT 1 - iT
good3 1−T2 1 - T^{2}
7 1−T2 1 - T^{2}
11 1+T2 1 + T^{2}
13 1−2iT−T2 1 - 2iT - T^{2}
19 1−T2 1 - T^{2}
23 1−T2 1 - T^{2}
29 1−T2 1 - T^{2}
31 1+T2 1 + T^{2}
37 1+T2 1 + T^{2}
41 1−T2 1 - T^{2}
43 1+T2 1 + T^{2}
47 1+T2 1 + T^{2}
53 1+2iT−T2 1 + 2iT - T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1+T2 1 + T^{2}
71 1+T2 1 + T^{2}
73 1+T2 1 + T^{2}
79 1+T2 1 + T^{2}
83 1+T2 1 + T^{2}
89 1−2T+T2 1 - 2T + T^{2}
97 1+T2 1 + 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

−9.550987724788604379431264379495, −8.947034852315499554544232547344, −8.109091838592266158893116795767, −7.07884851431177866454503796137, −6.29059528335442226075537437258, −5.06841425462184414006062935921, −4.21931866556954222613572583014, −3.70773392692290291612075294082, −2.22085192687463278883090077162, −1.48014408051926811139872589975, 0.948393339903121943391961748045, 2.84927817052814625843250387070, 3.90192539579700297486776828678, 4.87696887015863872538826880717, 5.53969064161163281918024913047, 6.41533775595560043029732049518, 7.43011918093119782203783332410, 7.69605804913797349617944069976, 8.685197224461549113258892412836, 9.493249164785248564425325113513

Graph of the ZZ-function along the critical line