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Properties

Label 2-21e2-63.34-c2-0-33
Degree 22
Conductor 441441
Sign 0.433−0.901i0.433 - 0.901i
Analytic cond. 12.016312.0163
Root an. cond. 3.466463.46646
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  + (0.662 − 1.14i)2-s + (2.99 − 0.0895i)3-s + (1.12 + 1.94i)4-s + (−6.26 + 3.61i)5-s + (1.88 − 3.49i)6-s + 8.27·8-s + (8.98 − 0.537i)9-s + 9.58i·10-s + (−8.56 + 14.8i)11-s + (3.54 + 5.72i)12-s + (−9.05 + 5.22i)13-s + (−18.4 + 11.4i)15-s + (0.990 − 1.71i)16-s + 6.23i·17-s + (5.33 − 10.6i)18-s − 11.8i·19-s + ⋯
L(s)  = 1  + (0.331 − 0.573i)2-s + (0.999 − 0.0298i)3-s + (0.280 + 0.486i)4-s + (−1.25 + 0.723i)5-s + (0.313 − 0.583i)6-s + 1.03·8-s + (0.998 − 0.0596i)9-s + 0.958i·10-s + (−0.778 + 1.34i)11-s + (0.295 + 0.477i)12-s + (−0.696 + 0.402i)13-s + (−1.23 + 0.760i)15-s + (0.0618 − 0.107i)16-s + 0.366i·17-s + (0.296 − 0.592i)18-s − 0.625i·19-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.433−0.901i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1) L(s)=((0.433−0.901i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.433−0.901i0.433 - 0.901i
Analytic conductor: 12.016312.0163
Root analytic conductor: 3.466463.46646
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ441(97,⋅)\chi_{441} (97, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1), 0.433−0.901i)(2,\ 441,\ (\ :1),\ 0.433 - 0.901i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.95866+1.23142i1.95866 + 1.23142i
L(12)L(\frac12) ≈\approx 1.95866+1.23142i1.95866 + 1.23142i
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)
bad3 1+(−2.99+0.0895i)T 1 + (-2.99 + 0.0895i)T
7 1 1
good2 1+(−0.662+1.14i)T+(−2−3.46i)T2 1 + (-0.662 + 1.14i)T + (-2 - 3.46i)T^{2}
5 1+(6.26−3.61i)T+(12.5−21.6i)T2 1 + (6.26 - 3.61i)T + (12.5 - 21.6i)T^{2}
11 1+(8.56−14.8i)T+(−60.5−104.i)T2 1 + (8.56 - 14.8i)T + (-60.5 - 104. i)T^{2}
13 1+(9.05−5.22i)T+(84.5−146.i)T2 1 + (9.05 - 5.22i)T + (84.5 - 146. i)T^{2}
17 1−6.23iT−289T2 1 - 6.23iT - 289T^{2}
19 1+11.8iT−361T2 1 + 11.8iT - 361T^{2}
23 1+(−5.85−10.1i)T+(−264.5+458.i)T2 1 + (-5.85 - 10.1i)T + (-264.5 + 458. i)T^{2}
29 1+(5.48−9.50i)T+(−420.5−728.i)T2 1 + (5.48 - 9.50i)T + (-420.5 - 728. i)T^{2}
31 1+(24.3−14.0i)T+(480.5−832.i)T2 1 + (24.3 - 14.0i)T + (480.5 - 832. i)T^{2}
37 1−67.4T+1.36e3T2 1 - 67.4T + 1.36e3T^{2}
41 1+(14.1−8.17i)T+(840.5−1.45e3i)T2 1 + (14.1 - 8.17i)T + (840.5 - 1.45e3i)T^{2}
43 1+(−31.2+54.2i)T+(−924.5−1.60e3i)T2 1 + (-31.2 + 54.2i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(−26.1−15.1i)T+(1.10e3+1.91e3i)T2 1 + (-26.1 - 15.1i)T + (1.10e3 + 1.91e3i)T^{2}
53 1−22.6T+2.80e3T2 1 - 22.6T + 2.80e3T^{2}
59 1+(1.58−0.916i)T+(1.74e3−3.01e3i)T2 1 + (1.58 - 0.916i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(−12.9−7.49i)T+(1.86e3+3.22e3i)T2 1 + (-12.9 - 7.49i)T + (1.86e3 + 3.22e3i)T^{2}
67 1+(29.7+51.4i)T+(−2.24e3+3.88e3i)T2 1 + (29.7 + 51.4i)T + (-2.24e3 + 3.88e3i)T^{2}
71 1−6.84T+5.04e3T2 1 - 6.84T + 5.04e3T^{2}
73 1+87.0iT−5.32e3T2 1 + 87.0iT - 5.32e3T^{2}
79 1+(34.3−59.5i)T+(−3.12e3−5.40e3i)T2 1 + (34.3 - 59.5i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(−30.2−17.4i)T+(3.44e3+5.96e3i)T2 1 + (-30.2 - 17.4i)T + (3.44e3 + 5.96e3i)T^{2}
89 1−14.9iT−7.92e3T2 1 - 14.9iT - 7.92e3T^{2}
97 1+(41.2+23.8i)T+(4.70e3+8.14e3i)T2 1 + (41.2 + 23.8i)T + (4.70e3 + 8.14e3i)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

−11.10372739623439412410281944486, −10.37487172209498045780532494167, −9.315004073046999332684073567327, −8.070928446054567240043161822255, −7.34539976851586879585465373791, −7.07601540296955579402458998539, −4.70551332037357589536713093145, −3.95120197889541891553461071712, −2.95815510627998880517416855513, −2.09930614407795447255560472607, 0.77054436202753780245554438931, 2.64953232420724279337392410267, 3.94364103226046977023889756614, 4.88127127444267681743288647825, 5.94552635635639242485588857243, 7.41036543342754087760178324424, 7.84746597697643610819295128658, 8.628210191308143367453459739746, 9.752303484067059652645650073945, 10.75766098765106905929596038691

Graph of the ZZ-function along the critical line