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Properties

Label 2-29e2-29.13-c1-0-51
Degree 22
Conductor 841841
Sign −0.553−0.832i-0.553 - 0.832i
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  + (0.702 − 1.45i)2-s + (−1.26 − 1.00i)3-s + (−0.385 − 0.483i)4-s + (−2.57 − 1.23i)5-s + (−2.35 + 1.13i)6-s + (1.39 − 1.74i)7-s + (2.18 − 0.497i)8-s + (−0.0849 − 0.372i)9-s + (−3.61 + 2.87i)10-s + (−3.52 − 0.805i)11-s + 1.00i·12-s + (0.942 − 4.12i)13-s + (−1.56 − 3.25i)14-s + (2.00 + 4.16i)15-s + (1.08 − 4.73i)16-s + 6.61i·17-s + ⋯
L(s)  = 1  + (0.496 − 1.03i)2-s + (−0.730 − 0.582i)3-s + (−0.192 − 0.241i)4-s + (−1.14 − 0.553i)5-s + (−0.962 + 0.463i)6-s + (0.526 − 0.660i)7-s + (0.770 − 0.175i)8-s + (−0.0283 − 0.124i)9-s + (−1.14 + 0.910i)10-s + (−1.06 − 0.242i)11-s + 0.288i·12-s + (0.261 − 1.14i)13-s + (−0.419 − 0.871i)14-s + (0.517 + 1.07i)15-s + (0.270 − 1.18i)16-s + 1.60i·17-s + ⋯

Functional equation

Λ(s)=(841s/2 ΓC(s) L(s)=((−0.553−0.832i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(841s/2 ΓC(s+1/2) L(s)=((−0.553−0.832i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 841841    =    29229^{2}
Sign: −0.553−0.832i-0.553 - 0.832i
Analytic conductor: 6.715416.71541
Root analytic conductor: 2.591412.59141
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ841(651,⋅)\chi_{841} (651, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 841, ( :1/2), −0.553−0.832i)(2,\ 841,\ (\ :1/2),\ -0.553 - 0.832i)

Particular Values

L(1)L(1) ≈\approx 0.426267+0.795633i0.426267 + 0.795633i
L(12)L(\frac12) ≈\approx 0.426267+0.795633i0.426267 + 0.795633i
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+(−0.702+1.45i)T+(−1.24−1.56i)T2 1 + (-0.702 + 1.45i)T + (-1.24 - 1.56i)T^{2}
3 1+(1.26+1.00i)T+(0.667+2.92i)T2 1 + (1.26 + 1.00i)T + (0.667 + 2.92i)T^{2}
5 1+(2.57+1.23i)T+(3.11+3.90i)T2 1 + (2.57 + 1.23i)T + (3.11 + 3.90i)T^{2}
7 1+(−1.39+1.74i)T+(−1.55−6.82i)T2 1 + (-1.39 + 1.74i)T + (-1.55 - 6.82i)T^{2}
11 1+(3.52+0.805i)T+(9.91+4.77i)T2 1 + (3.52 + 0.805i)T + (9.91 + 4.77i)T^{2}
13 1+(−0.942+4.12i)T+(−11.7−5.64i)T2 1 + (-0.942 + 4.12i)T + (-11.7 - 5.64i)T^{2}
17 1−6.61iT−17T2 1 - 6.61iT - 17T^{2}
19 1+(−1.44+1.15i)T+(4.22−18.5i)T2 1 + (-1.44 + 1.15i)T + (4.22 - 18.5i)T^{2}
23 1+(2.91−1.40i)T+(14.3−17.9i)T2 1 + (2.91 - 1.40i)T + (14.3 - 17.9i)T^{2}
31 1+(0.473−0.982i)T+(−19.3−24.2i)T2 1 + (0.473 - 0.982i)T + (-19.3 - 24.2i)T^{2}
37 1+(8.48−1.93i)T+(33.3−16.0i)T2 1 + (8.48 - 1.93i)T + (33.3 - 16.0i)T^{2}
41 1+2.85iT−41T2 1 + 2.85iT - 41T^{2}
43 1+(1.19+2.49i)T+(−26.8+33.6i)T2 1 + (1.19 + 2.49i)T + (-26.8 + 33.6i)T^{2}
47 1+(−6.82−1.55i)T+(42.3+20.3i)T2 1 + (-6.82 - 1.55i)T + (42.3 + 20.3i)T^{2}
53 1+(−1.80−0.867i)T+(33.0+41.4i)T2 1 + (-1.80 - 0.867i)T + (33.0 + 41.4i)T^{2}
59 1+5.09T+59T2 1 + 5.09T + 59T^{2}
61 1+(1.26+1.00i)T+(13.5+59.4i)T2 1 + (1.26 + 1.00i)T + (13.5 + 59.4i)T^{2}
67 1+(2.33+10.2i)T+(−60.3+29.0i)T2 1 + (2.33 + 10.2i)T + (-60.3 + 29.0i)T^{2}
71 1+(−0.339+1.48i)T+(−63.9−30.8i)T2 1 + (-0.339 + 1.48i)T + (-63.9 - 30.8i)T^{2}
73 1+(−0.126−0.262i)T+(−45.5+57.0i)T2 1 + (-0.126 - 0.262i)T + (-45.5 + 57.0i)T^{2}
79 1+(4.96−1.13i)T+(71.1−34.2i)T2 1 + (4.96 - 1.13i)T + (71.1 - 34.2i)T^{2}
83 1+(−4.95−6.21i)T+(−18.4+80.9i)T2 1 + (-4.95 - 6.21i)T + (-18.4 + 80.9i)T^{2}
89 1+(−3.77+7.84i)T+(−55.4−69.5i)T2 1 + (-3.77 + 7.84i)T + (-55.4 - 69.5i)T^{2}
97 1+(−12.9+10.3i)T+(21.5−94.5i)T2 1 + (-12.9 + 10.3i)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

−10.27093542373065654329610693818, −8.567514438542624997808575474724, −7.82699665260115446940160752637, −7.31135124162516674478192776900, −5.89651595437069971599711176366, −4.98407613907793877128240406886, −3.97760020827178767522547788574, −3.25127565560301820916385963631, −1.56928935141808877983371639217, −0.41539061033976021464472401644, 2.33096986653157171035170938966, 3.93058604844741250675204909030, 4.87738914011370768477976360317, 5.29853529150241532320659039234, 6.35668065614838897978663760420, 7.36468956214472346083682027704, 7.78266430533235991378141946298, 8.846451309759034450840125475590, 10.14985526883008795582614869255, 10.84128443867313087930849641866

Graph of the ZZ-function along the critical line