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Properties

Label 2-40e2-40.3-c1-0-13
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(543,⋅)\chi_{1600} (543, \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 1.5096457931.509645793
L(12)L(\frac12) ≈\approx 1.5096457931.509645793
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

−9.305393178039716966822555472399, −8.719766673044612478918708286302, −7.53445006873851160867671529588, −6.97740468308658975810006489468, −6.17265562401493949478572748374, −5.42856529628178976286610297630, −4.41173621166901332633662406709, −3.41250153430051590262418356790, −2.01597518886371727199565323716, −0.983405959453079691104987812441, 0.866450618518392576208124893054, 2.35349285089622839385830368836, 3.68069322451626743985511515298, 4.48253475421756990229897947276, 5.29255845273448946373582166915, 5.98821861704053599551446532266, 7.17467628510064910004177427988, 7.67933377654365724977655464813, 8.779328227780187777344334953433, 9.533014053654160072655803997924

Graph of the ZZ-function along the critical line