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Properties

Label 2-29e2-29.16-c1-0-43
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  + (2.17 + 1.04i)2-s + (1.50 − 1.88i)3-s + (2.38 + 2.99i)4-s + (0.900 + 0.433i)5-s + (5.25 − 2.52i)6-s + (−1.76 + 2.21i)7-s + (0.982 + 4.30i)8-s + (−0.629 − 2.75i)9-s + (1.50 + 1.88i)10-s + (0.0921 − 0.403i)11-s + 9.24·12-s + (0.851 − 3.73i)13-s + (−6.15 + 2.96i)14-s + (2.17 − 1.04i)15-s + (−0.667 + 2.92i)16-s + 0.828·17-s + ⋯
L(s)  = 1  + (1.53 + 0.740i)2-s + (0.869 − 1.08i)3-s + (1.19 + 1.49i)4-s + (0.402 + 0.194i)5-s + (2.14 − 1.03i)6-s + (−0.666 + 0.835i)7-s + (0.347 + 1.52i)8-s + (−0.209 − 0.919i)9-s + (0.475 + 0.596i)10-s + (0.0277 − 0.121i)11-s + 2.66·12-s + (0.236 − 1.03i)13-s + (−1.64 + 0.791i)14-s + (0.561 − 0.270i)15-s + (−0.166 + 0.731i)16-s + 0.200·17-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(190,⋅)\chi_{841} (190, \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 4.62519+1.07457i4.62519 + 1.07457i
L(12)L(\frac12) ≈\approx 4.62519+1.07457i4.62519 + 1.07457i
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+(−2.17−1.04i)T+(1.24+1.56i)T2 1 + (-2.17 - 1.04i)T + (1.24 + 1.56i)T^{2}
3 1+(−1.50+1.88i)T+(−0.667−2.92i)T2 1 + (-1.50 + 1.88i)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+(−0.0921+0.403i)T+(−9.91−4.77i)T2 1 + (-0.0921 + 0.403i)T + (-9.91 - 4.77i)T^{2}
13 1+(−0.851+3.73i)T+(−11.7−5.64i)T2 1 + (-0.851 + 3.73i)T + (-11.7 - 5.64i)T^{2}
17 1−0.828T+17T2 1 - 0.828T + 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+(3.29−1.58i)T+(14.3−17.9i)T2 1 + (3.29 - 1.58i)T + (14.3 - 17.9i)T^{2}
31 1+(9.07+4.36i)T+(19.3+24.2i)T2 1 + (9.07 + 4.36i)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+4.48T+41T2 1 + 4.48T + 41T^{2}
43 1+(3.23−1.55i)T+(26.8−33.6i)T2 1 + (3.23 - 1.55i)T + (26.8 - 33.6i)T^{2}
47 1+(−0.721+3.16i)T+(−42.3−20.3i)T2 1 + (-0.721 + 3.16i)T + (-42.3 - 20.3i)T^{2}
53 1+(8.54+4.11i)T+(33.0+41.4i)T2 1 + (8.54 + 4.11i)T + (33.0 + 41.4i)T^{2}
59 1+3.65T+59T2 1 + 3.65T + 59T^{2}
61 1+(3.01−3.77i)T+(−13.5−59.4i)T2 1 + (3.01 - 3.77i)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+(−1.96+8.60i)T+(−63.9−30.8i)T2 1 + (-1.96 + 8.60i)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+(−0.537−2.35i)T+(−71.1+34.2i)T2 1 + (-0.537 - 2.35i)T + (-71.1 + 34.2i)T^{2}
83 1+(−4.77−5.98i)T+(−18.4+80.9i)T2 1 + (-4.77 - 5.98i)T + (-18.4 + 80.9i)T^{2}
89 1+(−11.2−5.41i)T+(55.4+69.5i)T2 1 + (-11.2 - 5.41i)T + (55.4 + 69.5i)T^{2}
97 1+(−2.79−3.50i)T+(−21.5+94.5i)T2 1 + (-2.79 - 3.50i)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

−10.18566635862932477357101138028, −9.213168689471578030513790756434, −7.961891197116207900582142011151, −7.70785488256712084057939062605, −6.51969799423836953646575037183, −5.97476543282247582651906085788, −5.26437576886206323475558330964, −3.60835081971255033958820115165, −3.02964510995801303773866080692, −1.96607414579134336926880466710, 1.80488654398499356800401042462, 3.08809539918140425934697908220, 3.70490937671561437044981485088, 4.43911128322598169191444851060, 5.26328586233941450851222247546, 6.38095633382134218879814941976, 7.34749213611770537226741174760, 8.905346078584288620943674890727, 9.516375633684627774869056152274, 10.23928543007227661426759539969

Graph of the ZZ-function along the critical line