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Properties

Label 2-936-8.5-c1-0-23
Degree 22
Conductor 936936
Sign −0.707−0.707i-0.707 - 0.707i
Analytic cond. 7.473997.47399
Root an. cond. 2.733862.73386
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)2-s + 2i·4-s + 3i·5-s + 3·7-s + (−2 + 2i)8-s + (−3 + 3i)10-s − i·13-s + (3 + 3i)14-s − 4·16-s + 7·17-s + 4i·19-s − 6·20-s − 4·23-s − 4·25-s + (1 − i)26-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s + 1.34i·5-s + 1.13·7-s + (−0.707 + 0.707i)8-s + (−0.948 + 0.948i)10-s − 0.277i·13-s + (0.801 + 0.801i)14-s − 16-s + 1.69·17-s + 0.917i·19-s − 1.34·20-s − 0.834·23-s − 0.800·25-s + (0.196 − 0.196i)26-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((−0.707−0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((−0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: −0.707−0.707i-0.707 - 0.707i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(469,⋅)\chi_{936} (469, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), −0.707−0.707i)(2,\ 936,\ (\ :1/2),\ -0.707 - 0.707i)

Particular Values

L(1)L(1) ≈\approx 0.953463+2.30186i0.953463 + 2.30186i
L(12)L(\frac12) ≈\approx 0.953463+2.30186i0.953463 + 2.30186i
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−i)T 1 + (-1 - i)T
3 1 1
13 1+iT 1 + iT
good5 1−3iT−5T2 1 - 3iT - 5T^{2}
7 1−3T+7T2 1 - 3T + 7T^{2}
11 1−11T2 1 - 11T^{2}
17 1−7T+17T2 1 - 7T + 17T^{2}
19 1−4iT−19T2 1 - 4iT - 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+4iT−29T2 1 + 4iT - 29T^{2}
31 1+8T+31T2 1 + 8T + 31T^{2}
37 1+7iT−37T2 1 + 7iT - 37T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1−iT−43T2 1 - iT - 43T^{2}
47 1−7T+47T2 1 - 7T + 47T^{2}
53 1−4iT−53T2 1 - 4iT - 53T^{2}
59 1+14iT−59T2 1 + 14iT - 59T^{2}
61 1−10iT−61T2 1 - 10iT - 61T^{2}
67 1+2iT−67T2 1 + 2iT - 67T^{2}
71 1−3T+71T2 1 - 3T + 71T^{2}
73 1−14T+73T2 1 - 14T + 73T^{2}
79 1+10T+79T2 1 + 10T + 79T^{2}
83 1−14iT−83T2 1 - 14iT - 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1−8T+97T2 1 - 8T + 97T^{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

−10.54879325857465538070230369061, −9.568344033548428929447615449757, −8.189352302905368400678882268362, −7.74795126446030632420585194532, −7.04750305861168250111128132207, −5.90549497669378764643639949139, −5.41375683031812336635310272242, −4.06336662775372832913137138087, −3.30337636812120140812732157124, −2.07875543361596919789728301552, 1.01251428109686507359472261915, 1.91542847886084698137045959320, 3.45500436521338596356623048772, 4.54168481132733101164581341578, 5.12351612177070882626560888552, 5.80182269904285993340750573937, 7.23009806150085706545676071888, 8.256892312600616901149994718986, 9.002976634679524268556987485372, 9.810929708715935350486376354643

Graph of the ZZ-function along the critical line