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Properties

Label 2-2178-3.2-c2-0-3
Degree 22
Conductor 21782178
Sign −0.816+0.577i-0.816 + 0.577i
Analytic cond. 59.346259.3462
Root an. cond. 7.703647.70364
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 2.00·4-s + 9.68i·5-s − 8.70·7-s + 2.82i·8-s + 13.6·10-s − 2.23·13-s + 12.3i·14-s + 4.00·16-s + 6.99i·17-s − 6.91·19-s − 19.3i·20-s + 34.1i·23-s − 68.8·25-s + 3.15i·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s + 1.93i·5-s − 1.24·7-s + 0.353i·8-s + 1.36·10-s − 0.171·13-s + 0.879i·14-s + 0.250·16-s + 0.411i·17-s − 0.363·19-s − 0.968i·20-s + 1.48i·23-s − 2.75·25-s + 0.121i·26-s + ⋯

Functional equation

Λ(s)=(2178s/2 ΓC(s) L(s)=((−0.816+0.577i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(2178s/2 ΓC(s+1) L(s)=((−0.816+0.577i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21782178    =    2⋅32⋅1122 \cdot 3^{2} \cdot 11^{2}
Sign: −0.816+0.577i-0.816 + 0.577i
Analytic conductor: 59.346259.3462
Root analytic conductor: 7.703647.70364
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ2178(485,⋅)\chi_{2178} (485, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2178, ( :1), −0.816+0.577i)(2,\ 2178,\ (\ :1),\ -0.816 + 0.577i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.24732104440.2473210444
L(12)L(\frac12) ≈\approx 0.24732104440.2473210444
L(2)L(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+1.41iT 1 + 1.41iT
3 1 1
11 1 1
good5 1−9.68iT−25T2 1 - 9.68iT - 25T^{2}
7 1+8.70T+49T2 1 + 8.70T + 49T^{2}
13 1+2.23T+169T2 1 + 2.23T + 169T^{2}
17 1−6.99iT−289T2 1 - 6.99iT - 289T^{2}
19 1+6.91T+361T2 1 + 6.91T + 361T^{2}
23 1−34.1iT−529T2 1 - 34.1iT - 529T^{2}
29 1−48.7iT−841T2 1 - 48.7iT - 841T^{2}
31 1+7.91T+961T2 1 + 7.91T + 961T^{2}
37 1+22.1T+1.36e3T2 1 + 22.1T + 1.36e3T^{2}
41 1+64.4iT−1.68e3T2 1 + 64.4iT - 1.68e3T^{2}
43 1−60.7T+1.84e3T2 1 - 60.7T + 1.84e3T^{2}
47 1−53.8iT−2.20e3T2 1 - 53.8iT - 2.20e3T^{2}
53 1−37.7iT−2.80e3T2 1 - 37.7iT - 2.80e3T^{2}
59 1−46.6iT−3.48e3T2 1 - 46.6iT - 3.48e3T^{2}
61 1+25.2T+3.72e3T2 1 + 25.2T + 3.72e3T^{2}
67 1+15.8T+4.48e3T2 1 + 15.8T + 4.48e3T^{2}
71 1+29.9iT−5.04e3T2 1 + 29.9iT - 5.04e3T^{2}
73 1−32.4T+5.32e3T2 1 - 32.4T + 5.32e3T^{2}
79 1−23.4T+6.24e3T2 1 - 23.4T + 6.24e3T^{2}
83 1+44.7iT−6.88e3T2 1 + 44.7iT - 6.88e3T^{2}
89 1+118.iT−7.92e3T2 1 + 118. iT - 7.92e3T^{2}
97 1+157.T+9.40e3T2 1 + 157.T + 9.40e3T^{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

−9.551955958522009002308391944852, −8.865518684671621666112314004003, −7.49264639083782818183782502439, −7.13135466045663505430725674123, −6.21825918185623323619609878470, −5.60179434391594907256023802710, −4.06034703196046891392435224133, −3.32127545524278867183323799985, −2.87267307863490094008036961946, −1.79220137213635586703884392415, 0.079372996611393857147562001010, 0.820740660022706072552508742637, 2.35779806487288083796362643482, 3.79372372635846106985503168698, 4.51276934050306117474378160086, 5.23506949902301015754437741807, 6.08601336399113725452268812854, 6.70155784394622444819399158257, 7.86691267183561394932473568972, 8.372174603662678246966385476563

Graph of the ZZ-function along the critical line