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Properties

Label 2-3332-68.55-c0-0-1
Degree 22
Conductor 33323332
Sign −0.788+0.615i-0.788 + 0.615i
Analytic cond. 1.662881.66288
Root an. cond. 1.289521.28952
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  − i·2-s − 4-s + (1 − i)5-s + i·8-s − i·9-s + (−1 − i)10-s + 16-s + 17-s − 18-s + (−1 + i)20-s − i·25-s + (1 − i)29-s − i·32-s − i·34-s + i·36-s + (−1 + i)37-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (1 − i)5-s + i·8-s − i·9-s + (−1 − i)10-s + 16-s + 17-s − 18-s + (−1 + i)20-s − i·25-s + (1 − i)29-s − i·32-s − i·34-s + i·36-s + (−1 + i)37-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((−0.788+0.615i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((−0.788+0.615i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: −0.788+0.615i-0.788 + 0.615i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(2843,⋅)\chi_{3332} (2843, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), −0.788+0.615i)(2,\ 3332,\ (\ :0),\ -0.788 + 0.615i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3582322111.358232211
L(12)L(\frac12) ≈\approx 1.3582322111.358232211
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+iT 1 + iT
7 1 1
17 1−T 1 - T
good3 1+iT2 1 + iT^{2}
5 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
11 1−iT2 1 - iT^{2}
13 1+T2 1 + T^{2}
19 1+T2 1 + T^{2}
23 1−iT2 1 - iT^{2}
29 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
31 1+iT2 1 + iT^{2}
37 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
41 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
43 1+T2 1 + T^{2}
47 1−T2 1 - T^{2}
53 1−T2 1 - T^{2}
59 1+T2 1 + T^{2}
61 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
67 1−T2 1 - T^{2}
71 1+iT2 1 + iT^{2}
73 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
79 1−iT2 1 - iT^{2}
83 1+T2 1 + T^{2}
89 1+T2 1 + 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

−8.697110020316634563007092859195, −8.218461130640259915870701265046, −6.97185243947968735825637832093, −5.94295530227994839951083086231, −5.40497688836295314121031880538, −4.60127189344037187120258141629, −3.71589776636210439291140395299, −2.82000327023422200456686663783, −1.70607229695763933127210759749, −0.894378655439629435642484357070, 1.58963277607082267870183528224, 2.76639974321406096970467182040, 3.63283968892761488034441673367, 4.90901771012926610417118480275, 5.37516842637644558425263657366, 6.21055863605209663476274700992, 6.81343652147486035026172025044, 7.50575887639937940949158821573, 8.202334424155764257956765858584, 8.988317510774996668541736067437

Graph of the ZZ-function along the critical line