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Properties

Label 2-29e2-29.20-c1-0-31
Degree 22
Conductor 841841
Sign 0.897+0.440i0.897 + 0.440i
Analytic cond. 6.715416.71541
Root an. cond. 2.591412.59141
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  + (−0.373 + 0.179i)2-s + (−0.258 − 0.323i)3-s + (−1.14 + 1.42i)4-s + (0.900 − 0.433i)5-s + (0.154 + 0.0744i)6-s + (1.76 + 2.21i)7-s + (0.352 − 1.54i)8-s + (0.629 − 2.75i)9-s + (−0.258 + 0.323i)10-s + (−0.537 − 2.35i)11-s + 0.757·12-s + (−0.406 − 1.78i)13-s + (−1.05 − 0.508i)14-s + (−0.373 − 0.179i)15-s + (−0.667 − 2.92i)16-s − 4.82·17-s + ⋯
L(s)  = 1  + (−0.263 + 0.127i)2-s + (−0.149 − 0.186i)3-s + (−0.570 + 0.714i)4-s + (0.402 − 0.194i)5-s + (0.0631 + 0.0303i)6-s + (0.666 + 0.835i)7-s + (0.124 − 0.546i)8-s + (0.209 − 0.919i)9-s + (−0.0816 + 0.102i)10-s + (−0.161 − 0.709i)11-s + 0.218·12-s + (−0.112 − 0.494i)13-s + (−0.282 − 0.135i)14-s + (−0.0963 − 0.0464i)15-s + (−0.166 − 0.731i)16-s − 1.17·17-s + ⋯

Functional equation

Λ(s)=(841s/2 ΓC(s) L(s)=((0.897+0.440i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(841s/2 ΓC(s+1/2) L(s)=((0.897+0.440i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 841841    =    29229^{2}
Sign: 0.897+0.440i0.897 + 0.440i
Analytic conductor: 6.715416.71541
Root analytic conductor: 2.591412.59141
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ841(571,⋅)\chi_{841} (571, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 841, ( :1/2), 0.897+0.440i)(2,\ 841,\ (\ :1/2),\ 0.897 + 0.440i)

Particular Values

L(1)L(1) ≈\approx 1.19128−0.276771i1.19128 - 0.276771i
L(12)L(\frac12) ≈\approx 1.19128−0.276771i1.19128 - 0.276771i
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)
bad29 1 1
good2 1+(0.373−0.179i)T+(1.24−1.56i)T2 1 + (0.373 - 0.179i)T + (1.24 - 1.56i)T^{2}
3 1+(0.258+0.323i)T+(−0.667+2.92i)T2 1 + (0.258 + 0.323i)T + (-0.667 + 2.92i)T^{2}
5 1+(−0.900+0.433i)T+(3.11−3.90i)T2 1 + (-0.900 + 0.433i)T + (3.11 - 3.90i)T^{2}
7 1+(−1.76−2.21i)T+(−1.55+6.82i)T2 1 + (-1.76 - 2.21i)T + (-1.55 + 6.82i)T^{2}
11 1+(0.537+2.35i)T+(−9.91+4.77i)T2 1 + (0.537 + 2.35i)T + (-9.91 + 4.77i)T^{2}
13 1+(0.406+1.78i)T+(−11.7+5.64i)T2 1 + (0.406 + 1.78i)T + (-11.7 + 5.64i)T^{2}
17 1+4.82T+17T2 1 + 4.82T + 17T^{2}
19 1+(−3.74+4.69i)T+(−4.22−18.5i)T2 1 + (-3.74 + 4.69i)T + (-4.22 - 18.5i)T^{2}
23 1+(−6.89−3.32i)T+(14.3+17.9i)T2 1 + (-6.89 - 3.32i)T + (14.3 + 17.9i)T^{2}
31 1+(−3.66+1.76i)T+(19.3−24.2i)T2 1 + (-3.66 + 1.76i)T + (19.3 - 24.2i)T^{2}
37 1+(−0.890+3.89i)T+(−33.3−16.0i)T2 1 + (-0.890 + 3.89i)T + (-33.3 - 16.0i)T^{2}
41 1−12.4T+41T2 1 - 12.4T + 41T^{2}
43 1+(5.77+2.78i)T+(26.8+33.6i)T2 1 + (5.77 + 2.78i)T + (26.8 + 33.6i)T^{2}
47 1+(1.16+5.11i)T+(−42.3+20.3i)T2 1 + (1.16 + 5.11i)T + (-42.3 + 20.3i)T^{2}
53 1+(−6.74+3.24i)T+(33.0−41.4i)T2 1 + (-6.74 + 3.24i)T + (33.0 - 41.4i)T^{2}
59 1−7.65T+59T2 1 - 7.65T + 59T^{2}
61 1+(−0.516−0.647i)T+(−13.5+59.4i)T2 1 + (-0.516 - 0.647i)T + (-13.5 + 59.4i)T^{2}
67 1+(−1.25+5.51i)T+(−60.3−29.0i)T2 1 + (-1.25 + 5.51i)T + (-60.3 - 29.0i)T^{2}
71 1+(−0.705−3.09i)T+(−63.9+30.8i)T2 1 + (-0.705 - 3.09i)T + (-63.9 + 30.8i)T^{2}
73 1+(3.60+1.73i)T+(45.5+57.0i)T2 1 + (3.60 + 1.73i)T + (45.5 + 57.0i)T^{2}
79 1+(0.0921−0.403i)T+(−71.1−34.2i)T2 1 + (0.0921 - 0.403i)T + (-71.1 - 34.2i)T^{2}
83 1+(2.28−2.85i)T+(−18.4−80.9i)T2 1 + (2.28 - 2.85i)T + (-18.4 - 80.9i)T^{2}
89 1+(4.04−1.94i)T+(55.4−69.5i)T2 1 + (4.04 - 1.94i)T + (55.4 - 69.5i)T^{2}
97 1+(7.78−9.76i)T+(−21.5−94.5i)T2 1 + (7.78 - 9.76i)T + (-21.5 - 94.5i)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

−9.818932961463903689153006710391, −9.014505499256111663608859110819, −8.748850805193870397906731697995, −7.62241793973238226860682984167, −6.83628538504270198013161070106, −5.63948017770207630351029436899, −4.93010016344318944809037237290, −3.66901919993013094405462246209, −2.56996292500046844146149735898, −0.792948669759499669336840440010, 1.28218722185610260080672308533, 2.36580081380291123328599448126, 4.40210407870455286831331859621, 4.66192695467383077881124275538, 5.75768296274735442371573316212, 6.87581863788162857649625049384, 7.77065377149418489698354946394, 8.655917747233898463925237371638, 9.688660482120982748880229556752, 10.21888973929337115282511083160

Graph of the ZZ-function along the critical line