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Properties

Label 2-3822-13.12-c1-0-89
Degree 22
Conductor 38223822
Sign −0.832+0.554i-0.832 + 0.554i
Analytic cond. 30.518830.5188
Root an. cond. 5.524385.52438
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  + i·2-s − 3-s − 4-s − 2i·5-s − i·6-s − i·8-s + 9-s + 2·10-s + 12-s + (−2 − 3i)13-s + 2i·15-s + 16-s + 2·17-s + i·18-s − 4i·19-s + 2i·20-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.894i·5-s − 0.408i·6-s − 0.353i·8-s + 0.333·9-s + 0.632·10-s + 0.288·12-s + (−0.554 − 0.832i)13-s + 0.516i·15-s + 0.250·16-s + 0.485·17-s + 0.235i·18-s − 0.917i·19-s + 0.447i·20-s + ⋯

Functional equation

Λ(s)=(3822s/2 ΓC(s) L(s)=((−0.832+0.554i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3822s/2 ΓC(s+1/2) L(s)=((−0.832+0.554i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38223822    =    2⋅3⋅72⋅132 \cdot 3 \cdot 7^{2} \cdot 13
Sign: −0.832+0.554i-0.832 + 0.554i
Analytic conductor: 30.518830.5188
Root analytic conductor: 5.524385.52438
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3822(883,⋅)\chi_{3822} (883, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3822, ( :1/2), −0.832+0.554i)(2,\ 3822,\ (\ :1/2),\ -0.832 + 0.554i)

Particular Values

L(1)L(1) ≈\approx 0.39138041800.3913804180
L(12)L(\frac12) ≈\approx 0.39138041800.3913804180
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−iT 1 - iT
3 1+T 1 + T
7 1 1
13 1+(2+3i)T 1 + (2 + 3i)T
good5 1+2iT−5T2 1 + 2iT - 5T^{2}
11 1−11T2 1 - 11T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1+4iT−19T2 1 + 4iT - 19T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1−41T2 1 - 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+8iT−47T2 1 + 8iT - 47T^{2}
53 1−4T+53T2 1 - 4T + 53T^{2}
59 1−6iT−59T2 1 - 6iT - 59T^{2}
61 1+12T+61T2 1 + 12T + 61T^{2}
67 1+2iT−67T2 1 + 2iT - 67T^{2}
71 1−71T2 1 - 71T^{2}
73 1−14iT−73T2 1 - 14iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−14iT−83T2 1 - 14iT - 83T^{2}
89 1+4iT−89T2 1 + 4iT - 89T^{2}
97 1−2iT−97T2 1 - 2iT - 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

−8.122355185600567942613201423498, −7.45266808206281926041915776106, −6.73034273168710720939630488159, −5.78026219505288544744416246735, −5.33462430195331654459716392519, −4.62996282583798631118008252949, −3.84849602046665846154564978220, −2.57569096812133945889563895155, −1.14569926967265021163286808671, −0.13835932908597071675205284041, 1.43978161865337530302982254744, 2.36445560711683178088318063611, 3.31645978133578997846495246499, 4.13427384757802814094378207669, 4.89960585548359665993483001124, 5.92106320490358142633258051317, 6.42387833340555880189861123424, 7.39015759565747129628504855459, 7.930285637752151492286165146755, 8.998310390228690436984839026373

Graph of the ZZ-function along the critical line