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Properties

Label 2-2664-296.195-c0-0-2
Degree 22
Conductor 26642664
Sign 0.803+0.595i0.803 + 0.595i
Analytic cond. 1.329501.32950
Root an. cond. 1.153041.15304
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (1.22 − 0.707i)5-s + (−0.866 + 0.5i)7-s + (−0.707 + 0.707i)8-s + (−0.999 + i)10-s + 1.41·11-s + (−0.866 + 0.5i)13-s + (0.707 − 0.707i)14-s + (0.500 − 0.866i)16-s + (0.707 − 1.22i)17-s + (−1 − 1.73i)19-s + (0.707 − 1.22i)20-s + (−1.36 + 0.366i)22-s + (0.499 − 0.866i)25-s + (0.707 − 0.707i)26-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (1.22 − 0.707i)5-s + (−0.866 + 0.5i)7-s + (−0.707 + 0.707i)8-s + (−0.999 + i)10-s + 1.41·11-s + (−0.866 + 0.5i)13-s + (0.707 − 0.707i)14-s + (0.500 − 0.866i)16-s + (0.707 − 1.22i)17-s + (−1 − 1.73i)19-s + (0.707 − 1.22i)20-s + (−1.36 + 0.366i)22-s + (0.499 − 0.866i)25-s + (0.707 − 0.707i)26-s + ⋯

Functional equation

Λ(s)=(2664s/2 ΓC(s) L(s)=((0.803+0.595i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2664s/2 ΓC(s) L(s)=((0.803+0.595i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26642664    =    23⋅32⋅372^{3} \cdot 3^{2} \cdot 37
Sign: 0.803+0.595i0.803 + 0.595i
Analytic conductor: 1.329501.32950
Root analytic conductor: 1.153041.15304
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2664(1675,⋅)\chi_{2664} (1675, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2664, ( :0), 0.803+0.595i)(2,\ 2664,\ (\ :0),\ 0.803 + 0.595i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.91810046070.9181004607
L(12)L(\frac12) ≈\approx 0.91810046070.9181004607
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+(0.965−0.258i)T 1 + (0.965 - 0.258i)T
3 1 1
37 1+iT 1 + iT
good5 1+(−1.22+0.707i)T+(0.5−0.866i)T2 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}
7 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
11 1−1.41T+T2 1 - 1.41T + T^{2}
13 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
17 1+(−0.707+1.22i)T+(−0.5−0.866i)T2 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2}
19 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
23 1−T2 1 - T^{2}
29 1+1.41iT−T2 1 + 1.41iT - T^{2}
31 1−iT−T2 1 - iT - T^{2}
41 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
43 1−T+T2 1 - T + T^{2}
47 1−T2 1 - T^{2}
53 1+(−1.22−0.707i)T+(0.5+0.866i)T2 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}
59 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
71 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
73 1−T+T2 1 - T + T^{2}
79 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
83 1+(0.707−1.22i)T+(−0.5−0.866i)T2 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2}
89 1+(−0.707+1.22i)T+(−0.5−0.866i)T2 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)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

−9.124306103262496982109492962239, −8.618809233619038206752835727085, −7.24537505610714016671229594261, −6.77546161729457576628675267799, −6.02876532393127750265029763273, −5.36358351016700006355507944701, −4.37219272242696977367482007418, −2.76202061557499899899347122779, −2.15259673257433422354937282219, −0.867700864019320964787933752652, 1.38573769532661485801221645233, 2.21203150578460482511654761869, 3.34453537987329835192038173905, 3.92594159627016850352873045344, 5.68696814130685509270281141990, 6.32027793832515964834124796573, 6.72055904288276824384783638153, 7.63216535415811402654966215610, 8.475064075861241619417275992086, 9.345067508012580169395411474817

Graph of the ZZ-function along the critical line