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Properties

Label 2-825-11.9-c1-0-13
Degree 22
Conductor 825825
Sign −0.0536−0.998i-0.0536 - 0.998i
Analytic cond. 6.587656.58765
Root an. cond. 2.566642.56664
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  + (−0.150 + 0.464i)2-s + (−0.809 + 0.587i)3-s + (1.42 + 1.03i)4-s + (−0.150 − 0.464i)6-s + (4.13 + 3.00i)7-s + (−1.48 + 1.07i)8-s + (0.309 − 0.951i)9-s + (2.66 − 1.97i)11-s − 1.76·12-s + (−0.561 + 1.72i)13-s + (−2.01 + 1.46i)14-s + (0.811 + 2.49i)16-s + (−0.197 − 0.608i)17-s + (0.395 + 0.287i)18-s + (3.55 − 2.58i)19-s + ⋯
L(s)  = 1  + (−0.106 + 0.328i)2-s + (−0.467 + 0.339i)3-s + (0.712 + 0.517i)4-s + (−0.0616 − 0.189i)6-s + (1.56 + 1.13i)7-s + (−0.525 + 0.381i)8-s + (0.103 − 0.317i)9-s + (0.804 − 0.594i)11-s − 0.508·12-s + (−0.155 + 0.479i)13-s + (−0.539 + 0.391i)14-s + (0.202 + 0.624i)16-s + (−0.0479 − 0.147i)17-s + (0.0931 + 0.0676i)18-s + (0.816 − 0.593i)19-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=((−0.0536−0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0536 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s+1/2) L(s)=((−0.0536−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0536 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: −0.0536−0.998i-0.0536 - 0.998i
Analytic conductor: 6.587656.58765
Root analytic conductor: 2.566642.56664
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ825(526,⋅)\chi_{825} (526, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 825, ( :1/2), −0.0536−0.998i)(2,\ 825,\ (\ :1/2),\ -0.0536 - 0.998i)

Particular Values

L(1)L(1) ≈\approx 1.25447+1.32370i1.25447 + 1.32370i
L(12)L(\frac12) ≈\approx 1.25447+1.32370i1.25447 + 1.32370i
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)
bad3 1+(0.809−0.587i)T 1 + (0.809 - 0.587i)T
5 1 1
11 1+(−2.66+1.97i)T 1 + (-2.66 + 1.97i)T
good2 1+(0.150−0.464i)T+(−1.61−1.17i)T2 1 + (0.150 - 0.464i)T + (-1.61 - 1.17i)T^{2}
7 1+(−4.13−3.00i)T+(2.16+6.65i)T2 1 + (-4.13 - 3.00i)T + (2.16 + 6.65i)T^{2}
13 1+(0.561−1.72i)T+(−10.5−7.64i)T2 1 + (0.561 - 1.72i)T + (-10.5 - 7.64i)T^{2}
17 1+(0.197+0.608i)T+(−13.7+9.99i)T2 1 + (0.197 + 0.608i)T + (-13.7 + 9.99i)T^{2}
19 1+(−3.55+2.58i)T+(5.87−18.0i)T2 1 + (-3.55 + 2.58i)T + (5.87 - 18.0i)T^{2}
23 1−4.37T+23T2 1 - 4.37T + 23T^{2}
29 1+(6.11+4.44i)T+(8.96+27.5i)T2 1 + (6.11 + 4.44i)T + (8.96 + 27.5i)T^{2}
31 1+(−0.681+2.09i)T+(−25.0−18.2i)T2 1 + (-0.681 + 2.09i)T + (-25.0 - 18.2i)T^{2}
37 1+(4.83+3.51i)T+(11.4+35.1i)T2 1 + (4.83 + 3.51i)T + (11.4 + 35.1i)T^{2}
41 1+(−1.80+1.31i)T+(12.6−38.9i)T2 1 + (-1.80 + 1.31i)T + (12.6 - 38.9i)T^{2}
43 1+3.38T+43T2 1 + 3.38T + 43T^{2}
47 1+(−1.22+0.890i)T+(14.5−44.6i)T2 1 + (-1.22 + 0.890i)T + (14.5 - 44.6i)T^{2}
53 1+(2.83−8.72i)T+(−42.8−31.1i)T2 1 + (2.83 - 8.72i)T + (-42.8 - 31.1i)T^{2}
59 1+(4.34+3.15i)T+(18.2+56.1i)T2 1 + (4.34 + 3.15i)T + (18.2 + 56.1i)T^{2}
61 1+(−2.24−6.91i)T+(−49.3+35.8i)T2 1 + (-2.24 - 6.91i)T + (-49.3 + 35.8i)T^{2}
67 1+7.25T+67T2 1 + 7.25T + 67T^{2}
71 1+(−1.70−5.23i)T+(−57.4+41.7i)T2 1 + (-1.70 - 5.23i)T + (-57.4 + 41.7i)T^{2}
73 1+(5.77+4.19i)T+(22.5+69.4i)T2 1 + (5.77 + 4.19i)T + (22.5 + 69.4i)T^{2}
79 1+(−0.588+1.81i)T+(−63.9−46.4i)T2 1 + (-0.588 + 1.81i)T + (-63.9 - 46.4i)T^{2}
83 1+(0.0500+0.153i)T+(−67.1+48.7i)T2 1 + (0.0500 + 0.153i)T + (-67.1 + 48.7i)T^{2}
89 1+1.19T+89T2 1 + 1.19T + 89T^{2}
97 1+(−4.41+13.5i)T+(−78.4−57.0i)T2 1 + (-4.41 + 13.5i)T + (-78.4 - 57.0i)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

−10.79274333175780882050356594376, −9.233171684256154354338476381183, −8.843137491251926819155078801499, −7.84297993302844910360608796667, −7.06062347299629912122256315920, −5.96597736331411386995078717351, −5.32532189420997490060522123981, −4.23718079874961320851587882131, −2.87335349036795187618588466700, −1.66803558564197038692565353795, 1.15390620009759827987851334462, 1.77114962062176195418986647569, 3.48096610831853298440246496289, 4.77079027240403550377793086299, 5.48415791654095850597964164783, 6.76686077551023233781527911267, 7.28586556260057365985217744989, 8.093554419900376112514991683601, 9.375021931596445948551791851128, 10.34500874634034647964061340751

Graph of the ZZ-function along the critical line