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Properties

Label 2-704-88.21-c0-0-1
Degree 22
Conductor 704704
Sign −0.258−0.965i-0.258 - 0.965i
Analytic cond. 0.3513410.351341
Root an. cond. 0.5927400.592740
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·3-s + 1.73i·5-s − i·11-s − 1.73·15-s − 1.73·23-s − 1.99·25-s + i·27-s + 1.73·31-s + 33-s − 1.73i·37-s + 49-s + 1.73·55-s + i·59-s − i·67-s − 1.73i·69-s + ⋯
L(s)  = 1  + i·3-s + 1.73i·5-s − i·11-s − 1.73·15-s − 1.73·23-s − 1.99·25-s + i·27-s + 1.73·31-s + 33-s − 1.73i·37-s + 49-s + 1.73·55-s + i·59-s − i·67-s − 1.73i·69-s + ⋯

Functional equation

Λ(s)=(704s/2 ΓC(s) L(s)=((−0.258−0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(704s/2 ΓC(s) L(s)=((−0.258−0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 704704    =    26⋅112^{6} \cdot 11
Sign: −0.258−0.965i-0.258 - 0.965i
Analytic conductor: 0.3513410.351341
Root analytic conductor: 0.5927400.592740
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ704(417,⋅)\chi_{704} (417, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 704, ( :0), −0.258−0.965i)(2,\ 704,\ (\ :0),\ -0.258 - 0.965i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.94862252430.9486225243
L(12)L(\frac12) ≈\approx 0.94862252430.9486225243
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 1
11 1+iT 1 + iT
good3 1−iT−T2 1 - iT - T^{2}
5 1−1.73iT−T2 1 - 1.73iT - T^{2}
7 1−T2 1 - T^{2}
13 1+T2 1 + T^{2}
17 1−T2 1 - T^{2}
19 1+T2 1 + T^{2}
23 1+1.73T+T2 1 + 1.73T + T^{2}
29 1+T2 1 + T^{2}
31 1−1.73T+T2 1 - 1.73T + T^{2}
37 1+1.73iT−T2 1 + 1.73iT - T^{2}
41 1−T2 1 - T^{2}
43 1+T2 1 + T^{2}
47 1+T2 1 + T^{2}
53 1−T2 1 - T^{2}
59 1−iT−T2 1 - iT - T^{2}
61 1+T2 1 + T^{2}
67 1+iT−T2 1 + iT - T^{2}
71 1−1.73T+T2 1 - 1.73T + T^{2}
73 1−T2 1 - T^{2}
79 1−T2 1 - T^{2}
83 1+T2 1 + T^{2}
89 1+T+T2 1 + T + T^{2}
97 1+T+T2 1 + T + 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

−10.72484547208088205797295849686, −10.20745764341117358145841610966, −9.492945756502734424269664877455, −8.314266631517820958403245538905, −7.37671498738757632703418864884, −6.40521337497273557715870681164, −5.65899472335893004724788390613, −4.19158621722119490265550981881, −3.48722383147019187723139050942, −2.44568619602551735819792948858, 1.16728486179106102319791020140, 2.17212466483325350060575125605, 4.16609623914396943708179056102, 4.85720412065493504939454193658, 5.98962888256222667869782967086, 6.93407456699106966502233083049, 8.043359828562033382978911581842, 8.340571488010252642225880465354, 9.604008416826061200504921776161, 10.08040698691470360840492460271

Graph of the ZZ-function along the critical line