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Properties

Label 2-4400-5.4-c1-0-23
Degree 22
Conductor 44004400
Sign −0.894−0.447i-0.894 - 0.447i
Analytic cond. 35.134135.1341
Root an. cond. 5.927405.92740
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  + 2.79i·3-s − 0.208i·7-s − 4.79·9-s + 11-s − i·13-s − 0.791i·17-s + 6.58·19-s + 0.582·21-s + 3.79i·23-s − 4.99i·27-s − 6.79·29-s + 8.58·31-s + 2.79i·33-s + 2.58i·37-s + 2.79·39-s + ⋯
L(s)  = 1  + 1.61i·3-s − 0.0788i·7-s − 1.59·9-s + 0.301·11-s − 0.277i·13-s − 0.191i·17-s + 1.51·19-s + 0.127·21-s + 0.790i·23-s − 0.962i·27-s − 1.26·29-s + 1.54·31-s + 0.485i·33-s + 0.424i·37-s + 0.446·39-s + ⋯

Functional equation

Λ(s)=(4400s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4400s/2 ΓC(s+1/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 44004400    =    24⋅52⋅112^{4} \cdot 5^{2} \cdot 11
Sign: −0.894−0.447i-0.894 - 0.447i
Analytic conductor: 35.134135.1341
Root analytic conductor: 5.927405.92740
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4400(4049,⋅)\chi_{4400} (4049, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4400, ( :1/2), −0.894−0.447i)(2,\ 4400,\ (\ :1/2),\ -0.894 - 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.6607582331.660758233
L(12)L(\frac12) ≈\approx 1.6607582331.660758233
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
5 1 1
11 1−T 1 - T
good3 1−2.79iT−3T2 1 - 2.79iT - 3T^{2}
7 1+0.208iT−7T2 1 + 0.208iT - 7T^{2}
13 1+iT−13T2 1 + iT - 13T^{2}
17 1+0.791iT−17T2 1 + 0.791iT - 17T^{2}
19 1−6.58T+19T2 1 - 6.58T + 19T^{2}
23 1−3.79iT−23T2 1 - 3.79iT - 23T^{2}
29 1+6.79T+29T2 1 + 6.79T + 29T^{2}
31 1−8.58T+31T2 1 - 8.58T + 31T^{2}
37 1−2.58iT−37T2 1 - 2.58iT - 37T^{2}
41 1+1.41T+41T2 1 + 1.41T + 41T^{2}
43 1−10iT−43T2 1 - 10iT - 43T^{2}
47 1+1.41iT−47T2 1 + 1.41iT - 47T^{2}
53 1−11.3iT−53T2 1 - 11.3iT - 53T^{2}
59 1+10.5T+59T2 1 + 10.5T + 59T^{2}
61 1−4.20T+61T2 1 - 4.20T + 61T^{2}
67 1+4iT−67T2 1 + 4iT - 67T^{2}
71 1−10.7T+71T2 1 - 10.7T + 71T^{2}
73 1+7.79iT−73T2 1 + 7.79iT - 73T^{2}
79 1+15.5T+79T2 1 + 15.5T + 79T^{2}
83 1−9.95iT−83T2 1 - 9.95iT - 83T^{2}
89 1−0.791T+89T2 1 - 0.791T + 89T^{2}
97 1−6.20iT−97T2 1 - 6.20iT - 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

−8.947076518578473608542777152790, −7.994762833236819241757640961384, −7.37395342545581043961001092118, −6.26812650862192244829221147442, −5.50734546682872376662024379502, −4.91397410411789914065325994907, −4.16248956289390071705329839527, −3.41001153887090481038786686783, −2.77394191210548824842656234159, −1.19396089585866482458430720296, 0.51120410345282507646669023921, 1.47798096105292030401073893233, 2.30370194219875638942630951753, 3.19380512017666953124710520274, 4.23830960898633452174474143019, 5.39339243518740575403218658570, 5.93818089134249958150696063022, 6.89415877206793976574350537666, 7.11252583285465168772017647942, 8.001691636017243026483317449072

Graph of the ZZ-function along the critical line