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Properties

Label 2-3332-68.19-c0-0-1
Degree 22
Conductor 33323332
Sign −0.673−0.739i-0.673 - 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: −0.673−0.739i-0.673 - 0.739i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(2059,⋅)\chi_{3332} (2059, \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 2.1419049862.141904986
L(12)L(\frac12) ≈\approx 2.1419049862.141904986
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

−9.161511563119343540606500518318, −7.82562884592229213650325780499, −7.55424879790955622648732296622, −6.57032220503463674307292971235, −6.22901000901665781839665755070, −5.63198612189880790730157045095, −4.35716776850597121711549830772, −3.62707696532172022020501322927, −2.90474165750137057727457606987, −1.94827965416770219397599279817, 1.11342710516643518288797401479, 1.73998008984554966297433211330, 2.83461457848544409023271900029, 4.05975409333704124869233484045, 4.75249500590844874976955506936, 5.26899848961023729087418598942, 5.85304469049485541004284379128, 6.93437253246052987013893522412, 7.88717880989730488182322942049, 8.752627347151423413526107956341

Graph of the ZZ-function along the critical line