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Properties

Label 2-425-5.4-c3-0-29
Degree 22
Conductor 425425
Sign 0.894−0.447i0.894 - 0.447i
Analytic cond. 25.075825.0758
Root an. cond. 5.007575.00757
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.36i·2-s + 3.15i·3-s + 6.14·4-s + 4.29·6-s + 7.94i·7-s − 19.2i·8-s + 17.0·9-s + 27.6·11-s + 19.3i·12-s + 58.1i·13-s + 10.8·14-s + 22.9·16-s + 17i·17-s − 23.2i·18-s − 89.1·19-s + ⋯
L(s)  = 1  − 0.481i·2-s + 0.607i·3-s + 0.768·4-s + 0.292·6-s + 0.428i·7-s − 0.851i·8-s + 0.631·9-s + 0.756·11-s + 0.466i·12-s + 1.23i·13-s + 0.206·14-s + 0.358·16-s + 0.242i·17-s − 0.303i·18-s − 1.07·19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((0.894−0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+3/2) L(s)=((0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: 0.894−0.447i0.894 - 0.447i
Analytic conductor: 25.075825.0758
Root analytic conductor: 5.007575.00757
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ425(324,⋅)\chi_{425} (324, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :3/2), 0.894−0.447i)(2,\ 425,\ (\ :3/2),\ 0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 2.6364185362.636418536
L(12)L(\frac12) ≈\approx 2.6364185362.636418536
L(52)L(\frac{5}{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 1−17iT 1 - 17iT
good2 1+1.36iT−8T2 1 + 1.36iT - 8T^{2}
3 1−3.15iT−27T2 1 - 3.15iT - 27T^{2}
7 1−7.94iT−343T2 1 - 7.94iT - 343T^{2}
11 1−27.6T+1.33e3T2 1 - 27.6T + 1.33e3T^{2}
13 1−58.1iT−2.19e3T2 1 - 58.1iT - 2.19e3T^{2}
19 1+89.1T+6.85e3T2 1 + 89.1T + 6.85e3T^{2}
23 1+115.iT−1.21e4T2 1 + 115. iT - 1.21e4T^{2}
29 1−128.T+2.43e4T2 1 - 128.T + 2.43e4T^{2}
31 1−273.T+2.97e4T2 1 - 273.T + 2.97e4T^{2}
37 1−132.iT−5.06e4T2 1 - 132. iT - 5.06e4T^{2}
41 1+470.T+6.89e4T2 1 + 470.T + 6.89e4T^{2}
43 1−352.iT−7.95e4T2 1 - 352. iT - 7.95e4T^{2}
47 1+152.iT−1.03e5T2 1 + 152. iT - 1.03e5T^{2}
53 1−527.iT−1.48e5T2 1 - 527. iT - 1.48e5T^{2}
59 1−292.T+2.05e5T2 1 - 292.T + 2.05e5T^{2}
61 1+53.8T+2.26e5T2 1 + 53.8T + 2.26e5T^{2}
67 1+52.9iT−3.00e5T2 1 + 52.9iT - 3.00e5T^{2}
71 1−788.T+3.57e5T2 1 - 788.T + 3.57e5T^{2}
73 1−295.iT−3.89e5T2 1 - 295. iT - 3.89e5T^{2}
79 1−720.T+4.93e5T2 1 - 720.T + 4.93e5T^{2}
83 1+116.iT−5.71e5T2 1 + 116. iT - 5.71e5T^{2}
89 1−813.T+7.04e5T2 1 - 813.T + 7.04e5T^{2}
97 1+794.iT−9.12e5T2 1 + 794. iT - 9.12e5T^{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

−10.72308870657567844805290598227, −10.11629982777248473491691629807, −9.223626366010958020661931753624, −8.261910977690459780226048112589, −6.71165125949716709475273442944, −6.44961528817316771571179940816, −4.67928101884117464611030283685, −3.90021532882361645014123673841, −2.52850687783154344017929280299, −1.37657490741209906734996250300, 0.960096608816025830081263783952, 2.21727334428054659638299200789, 3.64420917737192377282506333751, 5.09055560433050006661600839449, 6.31443601514253187913459317505, 6.88966437912808954947995390043, 7.75663756959204948465481190503, 8.502704083337226475699960330998, 9.986759146171986558493080384504, 10.62749750277293710057057871285

Graph of the ZZ-function along the critical line