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Properties

Label 2-425-17.4-c1-0-17
Degree 22
Conductor 425425
Sign −0.615+0.788i-0.615 + 0.788i
Analytic cond. 3.393643.39364
Root an. cond. 1.842181.84218
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 + (−1 + i)3-s + 4-s + (1 + i)6-s + (−3 − 3i)7-s − 3i·8-s + i·9-s + (−3 − 3i)11-s + (−1 + i)12-s + (−3 + 3i)14-s − 16-s + (4 − i)17-s + 18-s − 6i·19-s + 6·21-s + (−3 + 3i)22-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.577 + 0.577i)3-s + 0.5·4-s + (0.408 + 0.408i)6-s + (−1.13 − 1.13i)7-s − 1.06i·8-s + 0.333i·9-s + (−0.904 − 0.904i)11-s + (−0.288 + 0.288i)12-s + (−0.801 + 0.801i)14-s − 0.250·16-s + (0.970 − 0.242i)17-s + 0.235·18-s − 1.37i·19-s + 1.30·21-s + (−0.639 + 0.639i)22-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((−0.615+0.788i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((−0.615+0.788i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: −0.615+0.788i-0.615 + 0.788i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(276,⋅)\chi_{425} (276, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), −0.615+0.788i)(2,\ 425,\ (\ :1/2),\ -0.615 + 0.788i)

Particular Values

L(1)L(1) ≈\approx 0.398624−0.816972i0.398624 - 0.816972i
L(12)L(\frac12) ≈\approx 0.398624−0.816972i0.398624 - 0.816972i
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)
bad5 1 1
17 1+(−4+i)T 1 + (-4 + i)T
good2 1+iT−2T2 1 + iT - 2T^{2}
3 1+(1−i)T−3iT2 1 + (1 - i)T - 3iT^{2}
7 1+(3+3i)T+7iT2 1 + (3 + 3i)T + 7iT^{2}
11 1+(3+3i)T+11iT2 1 + (3 + 3i)T + 11iT^{2}
13 1+13T2 1 + 13T^{2}
19 1+6iT−19T2 1 + 6iT - 19T^{2}
23 1+(1+i)T+23iT2 1 + (1 + i)T + 23iT^{2}
29 1+(−3+3i)T−29iT2 1 + (-3 + 3i)T - 29iT^{2}
31 1+(1−i)T−31iT2 1 + (1 - i)T - 31iT^{2}
37 1+(3−3i)T−37iT2 1 + (3 - 3i)T - 37iT^{2}
41 1+(3+3i)T+41iT2 1 + (3 + 3i)T + 41iT^{2}
43 1−12iT−43T2 1 - 12iT - 43T^{2}
47 1−2T+47T2 1 - 2T + 47T^{2}
53 1+2iT−53T2 1 + 2iT - 53T^{2}
59 1+6iT−59T2 1 + 6iT - 59T^{2}
61 1+(−1−i)T+61iT2 1 + (-1 - i)T + 61iT^{2}
67 1−6T+67T2 1 - 6T + 67T^{2}
71 1+(−3+3i)T−71iT2 1 + (-3 + 3i)T - 71iT^{2}
73 1+(−3+3i)T−73iT2 1 + (-3 + 3i)T - 73iT^{2}
79 1+(−7−7i)T+79iT2 1 + (-7 - 7i)T + 79iT^{2}
83 1−4iT−83T2 1 - 4iT - 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+(3−3i)T−97iT2 1 + (3 - 3i)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

−10.85288534787206863613664780116, −10.17638680979767677528936888510, −9.637382338936088951922805099041, −8.023010886338626353670450989520, −7.01810061752613472156754052628, −6.11787150208091906409812956620, −4.90685685004734445533013436417, −3.62961153618714600773447475715, −2.74580015880356677509674999510, −0.58750809406244438518557466809, 2.02401108121806693620180645918, 3.34266943572032839851220758708, 5.43728157632445237146996716750, 5.84959385588803731290290142881, 6.76714988496951488910683409710, 7.52990186095398680341139091394, 8.554667927277463650858942720792, 9.735353948737950290286309512124, 10.52651583056021977505408685280, 11.92351263353758655517547923465

Graph of the ZZ-function along the critical line