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Properties

Label 2-3332-68.43-c0-0-0
Degree 22
Conductor 33323332
Sign 0.673−0.739i0.673 - 0.739i
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  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 + 1.70i)5-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (0.707 + 1.70i)10-s − 1.00·16-s + (−0.707 + 0.707i)17-s + 1.00·18-s + (1.70 + 0.707i)20-s + (−1.70 − 1.70i)25-s + (−0.707 + 1.70i)29-s + (−0.707 + 0.707i)32-s + 1.00i·34-s + (0.707 − 0.707i)36-s + (1.70 + 0.707i)37-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 + 1.70i)5-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (0.707 + 1.70i)10-s − 1.00·16-s + (−0.707 + 0.707i)17-s + 1.00·18-s + (1.70 + 0.707i)20-s + (−1.70 − 1.70i)25-s + (−0.707 + 1.70i)29-s + (−0.707 + 0.707i)32-s + 1.00i·34-s + (0.707 − 0.707i)36-s + (1.70 + 0.707i)37-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((0.673−0.739i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((0.673−0.739i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 0.673−0.739i0.673 - 0.739i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(1471,⋅)\chi_{3332} (1471, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 0.673−0.739i)(2,\ 3332,\ (\ :0),\ 0.673 - 0.739i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3972667921.397266792
L(12)L(\frac12) ≈\approx 1.3972667921.397266792
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+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
7 1 1
17 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good3 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
5 1+(0.707−1.70i)T+(−0.707−0.707i)T2 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2}
11 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
13 1−T2 1 - T^{2}
19 1+iT2 1 + iT^{2}
23 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
29 1+(0.707−1.70i)T+(−0.707−0.707i)T2 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2}
31 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
37 1+(−1.70−0.707i)T+(0.707+0.707i)T2 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2}
41 1+(−0.292−0.707i)T+(−0.707+0.707i)T2 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2}
43 1−iT2 1 - iT^{2}
47 1+T2 1 + T^{2}
53 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
59 1−iT2 1 - iT^{2}
61 1+(−0.292−0.707i)T+(−0.707+0.707i)T2 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2}
67 1−T2 1 - T^{2}
71 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
73 1+(−0.707+1.70i)T+(−0.707−0.707i)T2 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2}
79 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
83 1+iT2 1 + iT^{2}
89 1+2iT−T2 1 + 2iT - T^{2}
97 1+(0.707−1.70i)T+(−0.707−0.707i)T2 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{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

−9.083023257289648643906607610005, −7.932501119867819205251752377147, −7.30165656548185722679899766640, −6.57899616628961079871496279015, −5.99954975419966063869477084708, −4.79402301021264576183173810456, −4.14120816936939216910316474815, −3.32725907116285632353854040213, −2.62417097887203082597183717696, −1.66724438166004028089889012926, 0.65721834447309321850082619888, 2.21442304092380673199566575360, 3.66746285278900886454033567017, 4.21932873224493985530803065026, 4.76753043535797753058362060340, 5.58335395566535234691631687726, 6.38074750558724713506387760046, 7.31417859732256799371939285686, 7.87631664110201393720062173909, 8.546195963377316710289372268751

Graph of the ZZ-function along the critical line