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Properties

Label 2-22e2-11.5-c1-0-0
Degree 22
Conductor 484484
Sign 0.437−0.899i0.437 - 0.899i
Analytic cond. 3.864753.86475
Root an. cond. 1.965891.96589
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  + (−0.809 − 0.587i)3-s + (−0.927 + 2.85i)5-s + (1.61 − 1.17i)7-s + (−0.618 − 1.90i)9-s + (1.23 + 3.80i)13-s + (2.42 − 1.76i)15-s + (−1.85 + 5.70i)17-s + (6.47 + 4.70i)19-s − 2·21-s − 3·23-s + (−3.23 − 2.35i)25-s + (−1.54 + 4.75i)27-s + (1.54 + 4.75i)31-s + (1.85 + 5.70i)35-s + (0.809 − 0.587i)37-s + ⋯
L(s)  = 1  + (−0.467 − 0.339i)3-s + (−0.414 + 1.27i)5-s + (0.611 − 0.444i)7-s + (−0.206 − 0.634i)9-s + (0.342 + 1.05i)13-s + (0.626 − 0.455i)15-s + (−0.449 + 1.38i)17-s + (1.48 + 1.07i)19-s − 0.436·21-s − 0.625·23-s + (−0.647 − 0.470i)25-s + (−0.297 + 0.915i)27-s + (0.277 + 0.854i)31-s + (0.313 + 0.964i)35-s + (0.133 − 0.0966i)37-s + ⋯

Functional equation

Λ(s)=(484s/2 ΓC(s) L(s)=((0.437−0.899i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(484s/2 ΓC(s+1/2) L(s)=((0.437−0.899i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 484484    =    22⋅1122^{2} \cdot 11^{2}
Sign: 0.437−0.899i0.437 - 0.899i
Analytic conductor: 3.864753.86475
Root analytic conductor: 1.965891.96589
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ484(269,⋅)\chi_{484} (269, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 484, ( :1/2), 0.437−0.899i)(2,\ 484,\ (\ :1/2),\ 0.437 - 0.899i)

Particular Values

L(1)L(1) ≈\approx 0.917858+0.573898i0.917858 + 0.573898i
L(12)L(\frac12) ≈\approx 0.917858+0.573898i0.917858 + 0.573898i
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)
bad2 1 1
11 1 1
good3 1+(0.809+0.587i)T+(0.927+2.85i)T2 1 + (0.809 + 0.587i)T + (0.927 + 2.85i)T^{2}
5 1+(0.927−2.85i)T+(−4.04−2.93i)T2 1 + (0.927 - 2.85i)T + (-4.04 - 2.93i)T^{2}
7 1+(−1.61+1.17i)T+(2.16−6.65i)T2 1 + (-1.61 + 1.17i)T + (2.16 - 6.65i)T^{2}
13 1+(−1.23−3.80i)T+(−10.5+7.64i)T2 1 + (-1.23 - 3.80i)T + (-10.5 + 7.64i)T^{2}
17 1+(1.85−5.70i)T+(−13.7−9.99i)T2 1 + (1.85 - 5.70i)T + (-13.7 - 9.99i)T^{2}
19 1+(−6.47−4.70i)T+(5.87+18.0i)T2 1 + (-6.47 - 4.70i)T + (5.87 + 18.0i)T^{2}
23 1+3T+23T2 1 + 3T + 23T^{2}
29 1+(8.96−27.5i)T2 1 + (8.96 - 27.5i)T^{2}
31 1+(−1.54−4.75i)T+(−25.0+18.2i)T2 1 + (-1.54 - 4.75i)T + (-25.0 + 18.2i)T^{2}
37 1+(−0.809+0.587i)T+(11.4−35.1i)T2 1 + (-0.809 + 0.587i)T + (11.4 - 35.1i)T^{2}
41 1+(12.6+38.9i)T2 1 + (12.6 + 38.9i)T^{2}
43 1−10T+43T2 1 - 10T + 43T^{2}
47 1+(14.5+44.6i)T2 1 + (14.5 + 44.6i)T^{2}
53 1+(1.85+5.70i)T+(−42.8+31.1i)T2 1 + (1.85 + 5.70i)T + (-42.8 + 31.1i)T^{2}
59 1+(2.42−1.76i)T+(18.2−56.1i)T2 1 + (2.42 - 1.76i)T + (18.2 - 56.1i)T^{2}
61 1+(−1.23+3.80i)T+(−49.3−35.8i)T2 1 + (-1.23 + 3.80i)T + (-49.3 - 35.8i)T^{2}
67 1+T+67T2 1 + T + 67T^{2}
71 1+(−4.63+14.2i)T+(−57.4−41.7i)T2 1 + (-4.63 + 14.2i)T + (-57.4 - 41.7i)T^{2}
73 1+(3.23−2.35i)T+(22.5−69.4i)T2 1 + (3.23 - 2.35i)T + (22.5 - 69.4i)T^{2}
79 1+(0.618+1.90i)T+(−63.9+46.4i)T2 1 + (0.618 + 1.90i)T + (-63.9 + 46.4i)T^{2}
83 1+(1.85−5.70i)T+(−67.1−48.7i)T2 1 + (1.85 - 5.70i)T + (-67.1 - 48.7i)T^{2}
89 1+9T+89T2 1 + 9T + 89T^{2}
97 1+(2.16+6.65i)T+(−78.4+57.0i)T2 1 + (2.16 + 6.65i)T + (-78.4 + 57.0i)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

−11.15679624540527634512232008309, −10.54294040774090417440630834271, −9.461710514557496028302263049568, −8.236239476541034713312465163456, −7.35282880578821108307230286617, −6.56518263730742926592386490519, −5.80503758436455896158296472628, −4.18844930802716877080359787905, −3.32869625136823597935180608635, −1.58062411332497186561083618198, 0.75657274691484477852318630432, 2.66317511789158502783548007910, 4.38964446451887250315531541308, 5.12014718249656613689065079337, 5.69060491927036264727887658014, 7.42396389834517857343184190716, 8.155017022335705648214371107394, 9.002696220187920954717621838272, 9.851736772060427031113255680687, 11.11871204488279129646591513574

Graph of the ZZ-function along the critical line