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Properties

Label 2-21e2-9.7-c1-0-33
Degree 22
Conductor 441441
Sign −0.904+0.426i-0.904 + 0.426i
Analytic cond. 3.521403.52140
Root an. cond. 1.876541.87654
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  + (1.23 − 2.13i)2-s + (1.73 − 0.0789i)3-s + (−2.02 − 3.51i)4-s + (−1.29 − 2.24i)5-s + (1.96 − 3.78i)6-s − 5.05·8-s + (2.98 − 0.273i)9-s − 6.38·10-s + (−2.25 + 3.90i)11-s + (−3.78 − 5.91i)12-s + (0.5 + 0.866i)13-s + (−2.42 − 3.78i)15-s + (−2.16 + 3.74i)16-s + 0.945·17-s + (3.09 − 6.70i)18-s + 4.05·19-s + ⋯
L(s)  = 1  + (0.869 − 1.50i)2-s + (0.998 − 0.0455i)3-s + (−1.01 − 1.75i)4-s + (−0.579 − 1.00i)5-s + (0.800 − 1.54i)6-s − 1.78·8-s + (0.995 − 0.0910i)9-s − 2.01·10-s + (−0.680 + 1.17i)11-s + (−1.09 − 1.70i)12-s + (0.138 + 0.240i)13-s + (−0.625 − 0.977i)15-s + (−0.540 + 0.936i)16-s + 0.229·17-s + (0.729 − 1.57i)18-s + 0.930·19-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((−0.904+0.426i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=((−0.904+0.426i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: −0.904+0.426i-0.904 + 0.426i
Analytic conductor: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ441(295,⋅)\chi_{441} (295, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), −0.904+0.426i)(2,\ 441,\ (\ :1/2),\ -0.904 + 0.426i)

Particular Values

L(1)L(1) ≈\approx 0.557773−2.49305i0.557773 - 2.49305i
L(12)L(\frac12) ≈\approx 0.557773−2.49305i0.557773 - 2.49305i
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)
bad3 1+(−1.73+0.0789i)T 1 + (-1.73 + 0.0789i)T
7 1 1
good2 1+(−1.23+2.13i)T+(−1−1.73i)T2 1 + (-1.23 + 2.13i)T + (-1 - 1.73i)T^{2}
5 1+(1.29+2.24i)T+(−2.5+4.33i)T2 1 + (1.29 + 2.24i)T + (-2.5 + 4.33i)T^{2}
11 1+(2.25−3.90i)T+(−5.5−9.52i)T2 1 + (2.25 - 3.90i)T + (-5.5 - 9.52i)T^{2}
13 1+(−0.5−0.866i)T+(−6.5+11.2i)T2 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2}
17 1−0.945T+17T2 1 - 0.945T + 17T^{2}
19 1−4.05T+19T2 1 - 4.05T + 19T^{2}
23 1+(−0.136−0.236i)T+(−11.5+19.9i)T2 1 + (-0.136 - 0.236i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.23−2.13i)T+(−14.5−25.1i)T2 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2}
31 1+(−1.16−2.01i)T+(−15.5+26.8i)T2 1 + (-1.16 - 2.01i)T + (-15.5 + 26.8i)T^{2}
37 1−1.78T+37T2 1 - 1.78T + 37T^{2}
41 1+(3.20+5.54i)T+(−20.5+35.5i)T2 1 + (3.20 + 5.54i)T + (-20.5 + 35.5i)T^{2}
43 1+(−5.21+9.03i)T+(−21.5−37.2i)T2 1 + (-5.21 + 9.03i)T + (-21.5 - 37.2i)T^{2}
47 1+(6.08−10.5i)T+(−23.5−40.7i)T2 1 + (6.08 - 10.5i)T + (-23.5 - 40.7i)T^{2}
53 1+6.27T+53T2 1 + 6.27T + 53T^{2}
59 1+(1.36+2.36i)T+(−29.5+51.0i)T2 1 + (1.36 + 2.36i)T + (-29.5 + 51.0i)T^{2}
61 1+(1.13−1.96i)T+(−30.5−52.8i)T2 1 + (1.13 - 1.96i)T + (-30.5 - 52.8i)T^{2}
67 1+(−7.90−13.6i)T+(−33.5+58.0i)T2 1 + (-7.90 - 13.6i)T + (-33.5 + 58.0i)T^{2}
71 1−3.27T+71T2 1 - 3.27T + 71T^{2}
73 1−1.50T+73T2 1 - 1.50T + 73T^{2}
79 1+(7.35−12.7i)T+(−39.5−68.4i)T2 1 + (7.35 - 12.7i)T + (-39.5 - 68.4i)T^{2}
83 1+(0.472−0.819i)T+(−41.5−71.8i)T2 1 + (0.472 - 0.819i)T + (-41.5 - 71.8i)T^{2}
89 1−14.3T+89T2 1 - 14.3T + 89T^{2}
97 1+(5.74−9.95i)T+(−48.5−84.0i)T2 1 + (5.74 - 9.95i)T + (-48.5 - 84.0i)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

−10.82909712009613245853448781677, −9.874436438657939238131871876322, −9.274138496809550614731158313637, −8.205276121267667167765027338139, −7.21849051213467371328397707365, −5.26602796065890056232445390093, −4.52668515029490494701086907385, −3.67382935062646437201913784312, −2.51461064959421322489554493286, −1.31601575231585731486460766373, 3.03253701240543513256816078408, 3.55713903744432719498868437644, 4.84132637571162115249113343366, 6.01344381641018650133733615465, 6.92830592506145766567673822144, 7.903645497640902363766965248304, 8.083128643324261488469836137875, 9.383435735684464530577169794374, 10.60299509757934506711681876132, 11.62783807590467910419495333404

Graph of the ZZ-function along the critical line