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Properties

Label 2-1734-17.13-c1-0-32
Degree 22
Conductor 17341734
Sign 0.638−0.769i0.638 - 0.769i
Analytic cond. 13.846013.8460
Root an. cond. 3.721023.72102
Motivic weight 11
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 + (−0.707 − 0.707i)3-s − 4-s + (2.82 + 2.82i)5-s + (0.707 − 0.707i)6-s + (1.41 − 1.41i)7-s − i·8-s + 1.00i·9-s + (−2.82 + 2.82i)10-s + (0.707 + 0.707i)12-s + 6·13-s + (1.41 + 1.41i)14-s − 4.00i·15-s + 16-s − 1.00·18-s − 4i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (1.26 + 1.26i)5-s + (0.288 − 0.288i)6-s + (0.534 − 0.534i)7-s − 0.353i·8-s + 0.333i·9-s + (−0.894 + 0.894i)10-s + (0.204 + 0.204i)12-s + 1.66·13-s + (0.377 + 0.377i)14-s − 1.03i·15-s + 0.250·16-s − 0.235·18-s − 0.917i·19-s + ⋯

Functional equation

Λ(s)=(1734s/2 ΓC(s) L(s)=((0.638−0.769i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1734s/2 ΓC(s+1/2) L(s)=((0.638−0.769i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17341734    =    2⋅3⋅1722 \cdot 3 \cdot 17^{2}
Sign: 0.638−0.769i0.638 - 0.769i
Analytic conductor: 13.846013.8460
Root analytic conductor: 3.721023.72102
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1734(829,⋅)\chi_{1734} (829, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1734, ( :1/2), 0.638−0.769i)(2,\ 1734,\ (\ :1/2),\ 0.638 - 0.769i)

Particular Values

L(1)L(1) ≈\approx 2.1187798782.118779878
L(12)L(\frac12) ≈\approx 2.1187798782.118779878
L(32)L(\frac{3}{2}) 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
3 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
17 1 1
good5 1+(−2.82−2.82i)T+5iT2 1 + (-2.82 - 2.82i)T + 5iT^{2}
7 1+(−1.41+1.41i)T−7iT2 1 + (-1.41 + 1.41i)T - 7iT^{2}
11 1−11iT2 1 - 11iT^{2}
13 1−6T+13T2 1 - 6T + 13T^{2}
19 1+4iT−19T2 1 + 4iT - 19T^{2}
23 1+(−4.24+4.24i)T−23iT2 1 + (-4.24 + 4.24i)T - 23iT^{2}
29 1+(−2.82−2.82i)T+29iT2 1 + (-2.82 - 2.82i)T + 29iT^{2}
31 1+(4.24+4.24i)T+31iT2 1 + (4.24 + 4.24i)T + 31iT^{2}
37 1+(2.82+2.82i)T+37iT2 1 + (2.82 + 2.82i)T + 37iT^{2}
41 1+(−7.07+7.07i)T−41iT2 1 + (-7.07 + 7.07i)T - 41iT^{2}
43 1+4iT−43T2 1 + 4iT - 43T^{2}
47 1+4T+47T2 1 + 4T + 47T^{2}
53 1−2iT−53T2 1 - 2iT - 53T^{2}
59 1−12iT−59T2 1 - 12iT - 59T^{2}
61 1+(−2.82+2.82i)T−61iT2 1 + (-2.82 + 2.82i)T - 61iT^{2}
67 1+12T+67T2 1 + 12T + 67T^{2}
71 1+(4.24+4.24i)T+71iT2 1 + (4.24 + 4.24i)T + 71iT^{2}
73 1+(1.41+1.41i)T+73iT2 1 + (1.41 + 1.41i)T + 73iT^{2}
79 1+(−7.07+7.07i)T−79iT2 1 + (-7.07 + 7.07i)T - 79iT^{2}
83 1−12iT−83T2 1 - 12iT - 83T^{2}
89 1−2T+89T2 1 - 2T + 89T^{2}
97 1+(4.24+4.24i)T+97iT2 1 + (4.24 + 4.24i)T + 97iT^{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

−9.234046011541761291967308079457, −8.686502994507477111413620248978, −7.45864331393173838062733459929, −6.99979779680319481542775363663, −6.20650363944950140985794417729, −5.76276538350345169889763579322, −4.71968002831761964485574728916, −3.51693811816339145109280867160, −2.34655988128672462293583320183, −1.11855930563323365891349631285, 1.18716447005332026001439736456, 1.74716469390857506638793077964, 3.21448737995741067852110916425, 4.32562108031942184072451296094, 5.15822150412737318690895925776, 5.69248826812880945939162036030, 6.39696254576639840421097014132, 8.092885864477195071309509803214, 8.678749113935464235375355975593, 9.284296117231742558428266471757

Graph of the ZZ-function along the critical line