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Properties

Label 2-29e2-29.4-c1-0-40
Degree 22
Conductor 841841
Sign −0.317+0.948i-0.317 + 0.948i
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  + (1.57 + 0.360i)2-s + (−0.702 − 1.45i)3-s + (0.556 + 0.268i)4-s + (−0.635 + 2.78i)5-s + (−0.582 − 2.55i)6-s + (−2.01 + 0.970i)7-s + (−1.74 − 1.39i)8-s + (0.238 − 0.298i)9-s + (−2.00 + 4.16i)10-s + (2.82 − 2.25i)11-s − 1.00i·12-s + (−2.64 − 3.31i)13-s + (−3.52 + 0.805i)14-s + (4.50 − 1.02i)15-s + (−3.02 − 3.79i)16-s − 6.61i·17-s + ⋯
L(s)  = 1  + (1.11 + 0.254i)2-s + (−0.405 − 0.841i)3-s + (0.278 + 0.134i)4-s + (−0.284 + 1.24i)5-s + (−0.237 − 1.04i)6-s + (−0.761 + 0.366i)7-s + (−0.618 − 0.492i)8-s + (0.0793 − 0.0995i)9-s + (−0.633 + 1.31i)10-s + (0.852 − 0.680i)11-s − 0.288i·12-s + (−0.732 − 0.918i)13-s + (−0.942 + 0.215i)14-s + (1.16 − 0.265i)15-s + (−0.756 − 0.948i)16-s − 1.60i·17-s + ⋯

Functional equation

Λ(s)=(841s/2 ΓC(s) L(s)=((−0.317+0.948i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(841s/2 ΓC(s+1/2) L(s)=((−0.317+0.948i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 841841    =    29229^{2}
Sign: −0.317+0.948i-0.317 + 0.948i
Analytic conductor: 6.715416.71541
Root analytic conductor: 2.591412.59141
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ841(236,⋅)\chi_{841} (236, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 841, ( :1/2), −0.317+0.948i)(2,\ 841,\ (\ :1/2),\ -0.317 + 0.948i)

Particular Values

L(1)L(1) ≈\approx 0.781326−1.08541i0.781326 - 1.08541i
L(12)L(\frac12) ≈\approx 0.781326−1.08541i0.781326 - 1.08541i
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+(−1.57−0.360i)T+(1.80+0.867i)T2 1 + (-1.57 - 0.360i)T + (1.80 + 0.867i)T^{2}
3 1+(0.702+1.45i)T+(−1.87+2.34i)T2 1 + (0.702 + 1.45i)T + (-1.87 + 2.34i)T^{2}
5 1+(0.635−2.78i)T+(−4.50−2.16i)T2 1 + (0.635 - 2.78i)T + (-4.50 - 2.16i)T^{2}
7 1+(2.01−0.970i)T+(4.36−5.47i)T2 1 + (2.01 - 0.970i)T + (4.36 - 5.47i)T^{2}
11 1+(−2.82+2.25i)T+(2.44−10.7i)T2 1 + (-2.82 + 2.25i)T + (2.44 - 10.7i)T^{2}
13 1+(2.64+3.31i)T+(−2.89+12.6i)T2 1 + (2.64 + 3.31i)T + (-2.89 + 12.6i)T^{2}
17 1+6.61iT−17T2 1 + 6.61iT - 17T^{2}
19 1+(−0.804+1.67i)T+(−11.8−14.8i)T2 1 + (-0.804 + 1.67i)T + (-11.8 - 14.8i)T^{2}
23 1+(0.720+3.15i)T+(−20.7+9.97i)T2 1 + (0.720 + 3.15i)T + (-20.7 + 9.97i)T^{2}
31 1+(1.06+0.242i)T+(27.9+13.4i)T2 1 + (1.06 + 0.242i)T + (27.9 + 13.4i)T^{2}
37 1+(−6.80−5.42i)T+(8.23+36.0i)T2 1 + (-6.80 - 5.42i)T + (8.23 + 36.0i)T^{2}
41 1−2.85iT−41T2 1 - 2.85iT - 41T^{2}
43 1+(2.69−0.615i)T+(38.7−18.6i)T2 1 + (2.69 - 0.615i)T + (38.7 - 18.6i)T^{2}
47 1+(5.47−4.36i)T+(10.4−45.8i)T2 1 + (5.47 - 4.36i)T + (10.4 - 45.8i)T^{2}
53 1+(−0.445+1.94i)T+(−47.7−22.9i)T2 1 + (-0.445 + 1.94i)T + (-47.7 - 22.9i)T^{2}
59 1+5.09T+59T2 1 + 5.09T + 59T^{2}
61 1+(0.702+1.45i)T+(−38.0+47.6i)T2 1 + (0.702 + 1.45i)T + (-38.0 + 47.6i)T^{2}
67 1+(−6.52+8.18i)T+(−14.9−65.3i)T2 1 + (-6.52 + 8.18i)T + (-14.9 - 65.3i)T^{2}
71 1+(0.952+1.19i)T+(−15.7+69.2i)T2 1 + (0.952 + 1.19i)T + (-15.7 + 69.2i)T^{2}
73 1+(−0.284+0.0649i)T+(65.7−31.6i)T2 1 + (-0.284 + 0.0649i)T + (65.7 - 31.6i)T^{2}
79 1+(−3.97−3.17i)T+(17.5+77.0i)T2 1 + (-3.97 - 3.17i)T + (17.5 + 77.0i)T^{2}
83 1+(7.15+3.44i)T+(51.7+64.8i)T2 1 + (7.15 + 3.44i)T + (51.7 + 64.8i)T^{2}
89 1+(−8.48−1.93i)T+(80.1+38.6i)T2 1 + (-8.48 - 1.93i)T + (80.1 + 38.6i)T^{2}
97 1+(−7.18+14.9i)T+(−60.4−75.8i)T2 1 + (-7.18 + 14.9i)T + (-60.4 - 75.8i)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

−9.865983282243302907604472220193, −9.317998346476779836653898950372, −7.80312517357243396344811633169, −6.84838735565543186282382360545, −6.53175762590985609633111809470, −5.76110836976910393864743615675, −4.63017688869680637464207446575, −3.26538715699956768298754027402, −2.84409863317501548258792133600, −0.47134792476444268192398113560, 1.83978937265980316183946182152, 3.77735304004766804104208061418, 4.10150200522458996014480493603, 4.84677681783079989248958755772, 5.67628632662782105716315843669, 6.69733002220172212439582917769, 7.961266521671800586905923798192, 9.090033971316476169399297262230, 9.573516128636878075545169310390, 10.48375713328716429810265154377

Graph of the ZZ-function along the critical line