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Properties

Label 2-54-27.23-c2-0-1
Degree 22
Conductor 5454
Sign −0.506−0.862i-0.506 - 0.862i
Analytic cond. 1.471391.47139
Root an. cond. 1.213011.21301
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  + (−0.483 + 1.32i)2-s + (−0.570 + 2.94i)3-s + (−1.53 − 1.28i)4-s + (3.98 + 0.702i)5-s + (−3.63 − 2.18i)6-s + (−10.1 + 8.49i)7-s + (2.44 − 1.41i)8-s + (−8.34 − 3.36i)9-s + (−2.86 + 4.95i)10-s + (13.3 − 2.35i)11-s + (4.66 − 3.77i)12-s + (17.4 − 6.34i)13-s + (−6.38 − 17.5i)14-s + (−4.34 + 11.3i)15-s + (0.694 + 3.93i)16-s + (13.8 + 8.01i)17-s + ⋯
L(s)  = 1  + (−0.241 + 0.664i)2-s + (−0.190 + 0.981i)3-s + (−0.383 − 0.321i)4-s + (0.797 + 0.140i)5-s + (−0.606 − 0.363i)6-s + (−1.44 + 1.21i)7-s + (0.306 − 0.176i)8-s + (−0.927 − 0.373i)9-s + (−0.286 + 0.495i)10-s + (1.21 − 0.214i)11-s + (0.388 − 0.314i)12-s + (1.34 − 0.488i)13-s + (−0.456 − 1.25i)14-s + (−0.289 + 0.756i)15-s + (0.0434 + 0.246i)16-s + (0.816 + 0.471i)17-s + ⋯

Functional equation

Λ(s)=(54s/2 ΓC(s) L(s)=((−0.506−0.862i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(54s/2 ΓC(s+1) L(s)=((−0.506−0.862i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5454    =    2⋅332 \cdot 3^{3}
Sign: −0.506−0.862i-0.506 - 0.862i
Analytic conductor: 1.471391.47139
Root analytic conductor: 1.213011.21301
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ54(23,⋅)\chi_{54} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 54, ( :1), −0.506−0.862i)(2,\ 54,\ (\ :1),\ -0.506 - 0.862i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.476736+0.833189i0.476736 + 0.833189i
L(12)L(\frac12) ≈\approx 0.476736+0.833189i0.476736 + 0.833189i
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)
bad2 1+(0.483−1.32i)T 1 + (0.483 - 1.32i)T
3 1+(0.570−2.94i)T 1 + (0.570 - 2.94i)T
good5 1+(−3.98−0.702i)T+(23.4+8.55i)T2 1 + (-3.98 - 0.702i)T + (23.4 + 8.55i)T^{2}
7 1+(10.1−8.49i)T+(8.50−48.2i)T2 1 + (10.1 - 8.49i)T + (8.50 - 48.2i)T^{2}
11 1+(−13.3+2.35i)T+(113.−41.3i)T2 1 + (-13.3 + 2.35i)T + (113. - 41.3i)T^{2}
13 1+(−17.4+6.34i)T+(129.−108.i)T2 1 + (-17.4 + 6.34i)T + (129. - 108. i)T^{2}
17 1+(−13.8−8.01i)T+(144.5+250.i)T2 1 + (-13.8 - 8.01i)T + (144.5 + 250. i)T^{2}
19 1+(−0.327−0.566i)T+(−180.5+312.i)T2 1 + (-0.327 - 0.566i)T + (-180.5 + 312. i)T^{2}
23 1+(−4.24+5.05i)T+(−91.8−520.i)T2 1 + (-4.24 + 5.05i)T + (-91.8 - 520. i)T^{2}
29 1+(−0.466+1.28i)T+(−644.−540.i)T2 1 + (-0.466 + 1.28i)T + (-644. - 540. i)T^{2}
31 1+(14.3+12.0i)T+(166.+946.i)T2 1 + (14.3 + 12.0i)T + (166. + 946. i)T^{2}
37 1+(8.43−14.6i)T+(−684.5−1.18e3i)T2 1 + (8.43 - 14.6i)T + (-684.5 - 1.18e3i)T^{2}
41 1+(14.6+40.2i)T+(−1.28e3+1.08e3i)T2 1 + (14.6 + 40.2i)T + (-1.28e3 + 1.08e3i)T^{2}
43 1+(−0.0113−0.0645i)T+(−1.73e3+632.i)T2 1 + (-0.0113 - 0.0645i)T + (-1.73e3 + 632. i)T^{2}
47 1+(−30.0−35.8i)T+(−383.+2.17e3i)T2 1 + (-30.0 - 35.8i)T + (-383. + 2.17e3i)T^{2}
53 1−14.3iT−2.80e3T2 1 - 14.3iT - 2.80e3T^{2}
59 1+(4.79+0.844i)T+(3.27e3+1.19e3i)T2 1 + (4.79 + 0.844i)T + (3.27e3 + 1.19e3i)T^{2}
61 1+(12.0−10.1i)T+(646.−3.66e3i)T2 1 + (12.0 - 10.1i)T + (646. - 3.66e3i)T^{2}
67 1+(37.1−13.5i)T+(3.43e3−2.88e3i)T2 1 + (37.1 - 13.5i)T + (3.43e3 - 2.88e3i)T^{2}
71 1+(60.9+35.1i)T+(2.52e3+4.36e3i)T2 1 + (60.9 + 35.1i)T + (2.52e3 + 4.36e3i)T^{2}
73 1+(34.1+59.1i)T+(−2.66e3+4.61e3i)T2 1 + (34.1 + 59.1i)T + (-2.66e3 + 4.61e3i)T^{2}
79 1+(−47.4−17.2i)T+(4.78e3+4.01e3i)T2 1 + (-47.4 - 17.2i)T + (4.78e3 + 4.01e3i)T^{2}
83 1+(−50.2+138.i)T+(−5.27e3−4.42e3i)T2 1 + (-50.2 + 138. i)T + (-5.27e3 - 4.42e3i)T^{2}
89 1+(17.2−9.98i)T+(3.96e3−6.85e3i)T2 1 + (17.2 - 9.98i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(−18.3−104.i)T+(−8.84e3+3.21e3i)T2 1 + (-18.3 - 104. i)T + (-8.84e3 + 3.21e3i)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

−15.65307975650822432801530542258, −14.69632027819863448680984770470, −13.47788069595621853071062588326, −12.07381275469583082843216979358, −10.44466501175413157190698438740, −9.414558328457004344990495284000, −8.757287012027398137909016332937, −6.20306429713994800271192291206, −5.82612214329808654653613878482, −3.49806634442314215538249746219, 1.25071308830012844554553736169, 3.56463644352525046976960997793, 6.13172776318420528214078018857, 7.15132024219230416899703673744, 9.001375871434019196588235088553, 10.01769694859273845319209684632, 11.34782534792233590028561040616, 12.58110461524586550875371538924, 13.52161848322369218281837179417, 14.01770488140292997348357037420

Graph of the ZZ-function along the critical line