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Properties

Label 2-22e2-11.3-c1-0-7
Degree 22
Conductor 484484
Sign −0.530+0.847i-0.530 + 0.847i
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 − 2.48i)3-s + (1.30 + 0.951i)5-s + (1.19 − 3.66i)7-s + (−3.11 + 2.26i)9-s + (1.92 − 1.40i)13-s + (1.30 − 4.02i)15-s + (1.92 + 1.40i)17-s + (−1.19 − 3.66i)19-s − 10.0·21-s − 2.47·23-s + (−0.736 − 2.26i)25-s + (1.80 + 1.31i)27-s + (−2.66 + 8.19i)29-s + (−0.690 + 0.502i)31-s + (5.04 − 3.66i)35-s + ⋯
L(s)  = 1  + (−0.467 − 1.43i)3-s + (0.585 + 0.425i)5-s + (0.450 − 1.38i)7-s + (−1.03 + 0.755i)9-s + (0.534 − 0.388i)13-s + (0.337 − 1.04i)15-s + (0.467 + 0.339i)17-s + (−0.273 − 0.840i)19-s − 2.20·21-s − 0.515·23-s + (−0.147 − 0.453i)25-s + (0.348 + 0.252i)27-s + (−0.494 + 1.52i)29-s + (−0.124 + 0.0901i)31-s + (0.852 − 0.619i)35-s + ⋯

Functional equation

Λ(s)=(484s/2 ΓC(s) L(s)=((−0.530+0.847i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(484s/2 ΓC(s+1/2) L(s)=((−0.530+0.847i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 484484    =    22⋅1122^{2} \cdot 11^{2}
Sign: −0.530+0.847i-0.530 + 0.847i
Analytic conductor: 3.864753.86475
Root analytic conductor: 1.965891.96589
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ484(245,⋅)\chi_{484} (245, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 484, ( :1/2), −0.530+0.847i)(2,\ 484,\ (\ :1/2),\ -0.530 + 0.847i)

Particular Values

L(1)L(1) ≈\approx 0.638521−1.15230i0.638521 - 1.15230i
L(12)L(\frac12) ≈\approx 0.638521−1.15230i0.638521 - 1.15230i
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+2.48i)T+(−2.42+1.76i)T2 1 + (0.809 + 2.48i)T + (-2.42 + 1.76i)T^{2}
5 1+(−1.30−0.951i)T+(1.54+4.75i)T2 1 + (-1.30 - 0.951i)T + (1.54 + 4.75i)T^{2}
7 1+(−1.19+3.66i)T+(−5.66−4.11i)T2 1 + (-1.19 + 3.66i)T + (-5.66 - 4.11i)T^{2}
13 1+(−1.92+1.40i)T+(4.01−12.3i)T2 1 + (-1.92 + 1.40i)T + (4.01 - 12.3i)T^{2}
17 1+(−1.92−1.40i)T+(5.25+16.1i)T2 1 + (-1.92 - 1.40i)T + (5.25 + 16.1i)T^{2}
19 1+(1.19+3.66i)T+(−15.3+11.1i)T2 1 + (1.19 + 3.66i)T + (-15.3 + 11.1i)T^{2}
23 1+2.47T+23T2 1 + 2.47T + 23T^{2}
29 1+(2.66−8.19i)T+(−23.4−17.0i)T2 1 + (2.66 - 8.19i)T + (-23.4 - 17.0i)T^{2}
31 1+(0.690−0.502i)T+(9.57−29.4i)T2 1 + (0.690 - 0.502i)T + (9.57 - 29.4i)T^{2}
37 1+(0.572−1.76i)T+(−29.9−21.7i)T2 1 + (0.572 - 1.76i)T + (-29.9 - 21.7i)T^{2}
41 1+(2.66+8.19i)T+(−33.1+24.0i)T2 1 + (2.66 + 8.19i)T + (-33.1 + 24.0i)T^{2}
43 1+43T2 1 + 43T^{2}
47 1+(−0.427−1.31i)T+(−38.0+27.6i)T2 1 + (-0.427 - 1.31i)T + (-38.0 + 27.6i)T^{2}
53 1+(−3.30+2.40i)T+(16.3−50.4i)T2 1 + (-3.30 + 2.40i)T + (16.3 - 50.4i)T^{2}
59 1+(−0.336+1.03i)T+(−47.7−34.6i)T2 1 + (-0.336 + 1.03i)T + (-47.7 - 34.6i)T^{2}
61 1+(−1.92−1.40i)T+(18.8+58.0i)T2 1 + (-1.92 - 1.40i)T + (18.8 + 58.0i)T^{2}
67 1−12.9T+67T2 1 - 12.9T + 67T^{2}
71 1+(−5.16−3.75i)T+(21.9+67.5i)T2 1 + (-5.16 - 3.75i)T + (21.9 + 67.5i)T^{2}
73 1+(−0.281+0.865i)T+(−59.0−42.9i)T2 1 + (-0.281 + 0.865i)T + (-59.0 - 42.9i)T^{2}
79 1+(5.78−4.20i)T+(24.4−75.1i)T2 1 + (5.78 - 4.20i)T + (24.4 - 75.1i)T^{2}
83 1+(−10.5−7.66i)T+(25.6+78.9i)T2 1 + (-10.5 - 7.66i)T + (25.6 + 78.9i)T^{2}
89 1−0.472T+89T2 1 - 0.472T + 89T^{2}
97 1+(−11.7+8.55i)T+(29.9−92.2i)T2 1 + (-11.7 + 8.55i)T + (29.9 - 92.2i)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

−10.75995237817235674501413115238, −10.11070645038356005015816054633, −8.621644805833393307978319747476, −7.65778654935867587374017488855, −6.99020009699597258661366239713, −6.28391953474641100465568500152, −5.21637355879491133525905772197, −3.69800636310515816265694489778, −2.05828146868527219068607943632, −0.890890576571586696560683311209, 2.03054815940115123780382073172, 3.63780121958315386093209429462, 4.74396113408630387427084492874, 5.59140007141966739848062126577, 6.09398667100770180032193753771, 7.971831545909616965577349615314, 8.915274851079268624966294248502, 9.553300732954448761024702264402, 10.21260498023722732551512060975, 11.36889184451215961412180185360

Graph of the ZZ-function along the critical line