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Properties

Label 2-1440-5.2-c0-0-2
Degree 22
Conductor 14401440
Sign 0.973+0.229i0.973 + 0.229i
Analytic cond. 0.7186530.718653
Root an. cond. 0.8477340.847734
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  + 5-s + (1 − i)13-s + (−1 − i)17-s + 25-s + 2i·29-s + (1 + i)37-s − 2·41-s − i·49-s + (1 − i)53-s + (1 − i)65-s + (−1 + i)73-s + (−1 − i)85-s + 2i·89-s + (−1 − i)97-s + 2i·109-s + ⋯
L(s)  = 1  + 5-s + (1 − i)13-s + (−1 − i)17-s + 25-s + 2i·29-s + (1 + i)37-s − 2·41-s − i·49-s + (1 − i)53-s + (1 − i)65-s + (−1 + i)73-s + (−1 − i)85-s + 2i·89-s + (−1 − i)97-s + 2i·109-s + ⋯

Functional equation

Λ(s)=(1440s/2 ΓC(s) L(s)=((0.973+0.229i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1440s/2 ΓC(s) L(s)=((0.973+0.229i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14401440    =    25⋅32⋅52^{5} \cdot 3^{2} \cdot 5
Sign: 0.973+0.229i0.973 + 0.229i
Analytic conductor: 0.7186530.718653
Root analytic conductor: 0.8477340.847734
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1440(577,⋅)\chi_{1440} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1440, ( :0), 0.973+0.229i)(2,\ 1440,\ (\ :0),\ 0.973 + 0.229i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3157316241.315731624
L(12)L(\frac12) ≈\approx 1.3157316241.315731624
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 1
3 1 1
5 1−T 1 - T
good7 1+iT2 1 + iT^{2}
11 1+T2 1 + T^{2}
13 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
17 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
19 1−T2 1 - T^{2}
23 1−iT2 1 - iT^{2}
29 1−2iT−T2 1 - 2iT - T^{2}
31 1+T2 1 + T^{2}
37 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
41 1+2T+T2 1 + 2T + T^{2}
43 1−iT2 1 - iT^{2}
47 1+iT2 1 + iT^{2}
53 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1+iT2 1 + iT^{2}
71 1+T2 1 + T^{2}
73 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
79 1−T2 1 - T^{2}
83 1−iT2 1 - iT^{2}
89 1−2iT−T2 1 - 2iT - T^{2}
97 1+(1+i)T+iT2 1 + (1 + i)T + iT^{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

−9.750468171704631929740761426681, −8.819825118749431193299193415884, −8.356620180901545215757275275054, −7.04522918523971434102280963918, −6.50447129242142655992783928096, −5.47128246144532710457912472558, −4.91304121345842909654633126375, −3.51573612738716185265442853736, −2.59686168500782016158249376578, −1.32639248226309068763830315054, 1.57524917341929851110694534915, 2.45924310097564594588447621042, 3.87085004248764801643048610709, 4.63430813009946904863747690975, 5.99698087262425325432497810660, 6.19668732674116076915441533266, 7.22003887251406042531982736089, 8.379783667186916001121314056584, 8.958377295751747429461176436404, 9.704891110795053292508302904530

Graph of the ZZ-function along the critical line