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Properties

Label 2-40e2-40.27-c1-0-14
Degree 22
Conductor 16001600
Sign 0.973−0.229i0.973 - 0.229i
Analytic cond. 12.776012.7760
Root an. cond. 3.574363.57436
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  + (1 − i)3-s + (1 − i)7-s + i·9-s − 4·11-s + (3 + 3i)13-s + (3 + 3i)17-s + 6i·19-s − 2i·21-s + (3 + 3i)23-s + (4 + 4i)27-s + 2·29-s − 6i·31-s + (−4 + 4i)33-s + (3 − 3i)37-s + 6·39-s + ⋯
L(s)  = 1  + (0.577 − 0.577i)3-s + (0.377 − 0.377i)7-s + 0.333i·9-s − 1.20·11-s + (0.832 + 0.832i)13-s + (0.727 + 0.727i)17-s + 1.37i·19-s − 0.436i·21-s + (0.625 + 0.625i)23-s + (0.769 + 0.769i)27-s + 0.371·29-s − 1.07i·31-s + (−0.696 + 0.696i)33-s + (0.493 − 0.493i)37-s + 0.960·39-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((0.973−0.229i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+1/2) L(s)=((0.973−0.229i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 0.973−0.229i0.973 - 0.229i
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1600(607,⋅)\chi_{1600} (607, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1600, ( :1/2), 0.973−0.229i)(2,\ 1600,\ (\ :1/2),\ 0.973 - 0.229i)

Particular Values

L(1)L(1) ≈\approx 2.1349615552.134961555
L(12)L(\frac12) ≈\approx 2.1349615552.134961555
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 1
5 1 1
good3 1+(−1+i)T−3iT2 1 + (-1 + i)T - 3iT^{2}
7 1+(−1+i)T−7iT2 1 + (-1 + i)T - 7iT^{2}
11 1+4T+11T2 1 + 4T + 11T^{2}
13 1+(−3−3i)T+13iT2 1 + (-3 - 3i)T + 13iT^{2}
17 1+(−3−3i)T+17iT2 1 + (-3 - 3i)T + 17iT^{2}
19 1−6iT−19T2 1 - 6iT - 19T^{2}
23 1+(−3−3i)T+23iT2 1 + (-3 - 3i)T + 23iT^{2}
29 1−2T+29T2 1 - 2T + 29T^{2}
31 1+6iT−31T2 1 + 6iT - 31T^{2}
37 1+(−3+3i)T−37iT2 1 + (-3 + 3i)T - 37iT^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1+(3−3i)T−43iT2 1 + (3 - 3i)T - 43iT^{2}
47 1+(−9+9i)T−47iT2 1 + (-9 + 9i)T - 47iT^{2}
53 1+(5+5i)T+53iT2 1 + (5 + 5i)T + 53iT^{2}
59 1+10iT−59T2 1 + 10iT - 59T^{2}
61 1−12iT−61T2 1 - 12iT - 61T^{2}
67 1+(9+9i)T+67iT2 1 + (9 + 9i)T + 67iT^{2}
71 1+6iT−71T2 1 + 6iT - 71T^{2}
73 1+(5−5i)T−73iT2 1 + (5 - 5i)T - 73iT^{2}
79 1+79T2 1 + 79T^{2}
83 1+(3−3i)T−83iT2 1 + (3 - 3i)T - 83iT^{2}
89 1−89T2 1 - 89T^{2}
97 1+(−7−7i)T+97iT2 1 + (-7 - 7i)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

−9.335471871715548963134810547260, −8.379833289083604216675323831739, −7.84634544188999616789983521419, −7.37277993114567478708472277995, −6.18081979363828298268674060771, −5.42686353838499235316846106494, −4.32105652789930491658332492539, −3.37673221799184133562580383159, −2.22009061319121592116103461872, −1.32389948059462705641449834747, 0.868681328818889429323508678969, 2.76498067237927063748406916396, 3.04649248551751460664454761990, 4.42221592988129295533729913592, 5.14303012029710053545229930093, 6.00074081911287784179597347611, 7.11121889197624676284593511399, 7.957129164992029050136585001242, 8.710792514662502558128758756072, 9.199986036096361040644173445779

Graph of the ZZ-function along the critical line