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Properties

Label 2-825-15.8-c1-0-14
Degree 22
Conductor 825825
Sign −0.229+0.973i-0.229 + 0.973i
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  + (−1.22 − 1.22i)2-s + (−1.22 − 1.22i)3-s + 0.999i·4-s + 2.99i·6-s + (−2.44 + 2.44i)7-s + (−1.22 + 1.22i)8-s + 2.99i·9-s + i·11-s + (1.22 − 1.22i)12-s + (2.44 + 2.44i)13-s + 5.99·14-s + 5·16-s + (−4.89 − 4.89i)17-s + (3.67 − 3.67i)18-s − 2i·19-s + ⋯
L(s)  = 1  + (−0.866 − 0.866i)2-s + (−0.707 − 0.707i)3-s + 0.499i·4-s + 1.22i·6-s + (−0.925 + 0.925i)7-s + (−0.433 + 0.433i)8-s + 0.999i·9-s + 0.301i·11-s + (0.353 − 0.353i)12-s + (0.679 + 0.679i)13-s + 1.60·14-s + 1.25·16-s + (−1.18 − 1.18i)17-s + (0.866 − 0.866i)18-s − 0.458i·19-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=((−0.229+0.973i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s+1/2) L(s)=((−0.229+0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: −0.229+0.973i-0.229 + 0.973i
Analytic conductor: 6.587656.58765
Root analytic conductor: 2.566642.56664
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ825(518,⋅)\chi_{825} (518, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 825, ( :1/2), −0.229+0.973i)(2,\ 825,\ (\ :1/2),\ -0.229 + 0.973i)

Particular Values

L(1)L(1) ≈\approx 0.328579−0.415178i0.328579 - 0.415178i
L(12)L(\frac12) ≈\approx 0.328579−0.415178i0.328579 - 0.415178i
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+(1.22+1.22i)T 1 + (1.22 + 1.22i)T
5 1 1
11 1−iT 1 - iT
good2 1+(1.22+1.22i)T+2iT2 1 + (1.22 + 1.22i)T + 2iT^{2}
7 1+(2.44−2.44i)T−7iT2 1 + (2.44 - 2.44i)T - 7iT^{2}
13 1+(−2.44−2.44i)T+13iT2 1 + (-2.44 - 2.44i)T + 13iT^{2}
17 1+(4.89+4.89i)T+17iT2 1 + (4.89 + 4.89i)T + 17iT^{2}
19 1+2iT−19T2 1 + 2iT - 19T^{2}
23 1+(−4.89+4.89i)T−23iT2 1 + (-4.89 + 4.89i)T - 23iT^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1−37iT2 1 - 37iT^{2}
41 1+6iT−41T2 1 + 6iT - 41T^{2}
43 1+(−7.34−7.34i)T+43iT2 1 + (-7.34 - 7.34i)T + 43iT^{2}
47 1+(−4.89−4.89i)T+47iT2 1 + (-4.89 - 4.89i)T + 47iT^{2}
53 1+(−4.89+4.89i)T−53iT2 1 + (-4.89 + 4.89i)T - 53iT^{2}
59 1+59T2 1 + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1+(2.44−2.44i)T−67iT2 1 + (2.44 - 2.44i)T - 67iT^{2}
71 1−12iT−71T2 1 - 12iT - 71T^{2}
73 1+(2.44+2.44i)T+73iT2 1 + (2.44 + 2.44i)T + 73iT^{2}
79 1+10iT−79T2 1 + 10iT - 79T^{2}
83 1+(−7.34+7.34i)T−83iT2 1 + (-7.34 + 7.34i)T - 83iT^{2}
89 1−12T+89T2 1 - 12T + 89T^{2}
97 1+(−9.79+9.79i)T−97iT2 1 + (-9.79 + 9.79i)T - 97iT^{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

−10.01368933805154981693588478598, −9.010982543903517233195043798873, −8.838323093869874563040642904431, −7.37054315867894742638343854616, −6.52341874887124592549351828444, −5.78296719568389731010845163604, −4.64232924007728050199956998151, −2.83110845861132922722951454565, −2.11547747001274555263406051458, −0.59891573596854514569297922697, 0.77848454641705121617388825512, 3.42429216389021605845858533824, 3.99890462760827521878461780204, 5.55690593678711354309153587803, 6.27769627525307371960362385828, 6.93668977679232624864165114260, 7.898102002711929811285412962623, 8.922055394475718171978809646328, 9.457550556024299715679062740995, 10.51332853786189906090162755281

Graph of the ZZ-function along the critical line