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Properties

Label 2-252-9.7-c1-0-3
Degree 22
Conductor 252252
Sign 0.998+0.0576i0.998 + 0.0576i
Analytic cond. 2.012232.01223
Root an. cond. 1.418531.41853
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  + (1.64 − 0.545i)3-s + (0.849 + 1.47i)5-s + (0.5 − 0.866i)7-s + (2.40 − 1.79i)9-s + (−1.23 + 2.14i)11-s + (−0.388 − 0.673i)13-s + (2.19 + 1.95i)15-s + 2.81·17-s − 4.98·19-s + (0.349 − 1.69i)21-s + (−0.356 − 0.616i)23-s + (1.05 − 1.82i)25-s + (2.97 − 4.25i)27-s + (−2.25 + 3.90i)29-s + (−2.54 − 4.41i)31-s + ⋯
L(s)  = 1  + (0.949 − 0.314i)3-s + (0.380 + 0.658i)5-s + (0.188 − 0.327i)7-s + (0.801 − 0.597i)9-s + (−0.373 + 0.646i)11-s + (−0.107 − 0.186i)13-s + (0.567 + 0.505i)15-s + 0.681·17-s − 1.14·19-s + (0.0763 − 0.370i)21-s + (−0.0742 − 0.128i)23-s + (0.211 − 0.365i)25-s + (0.572 − 0.819i)27-s + (−0.418 + 0.725i)29-s + (−0.457 − 0.793i)31-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=((0.998+0.0576i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0576i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+1/2) L(s)=((0.998+0.0576i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0576i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 0.998+0.0576i0.998 + 0.0576i
Analytic conductor: 2.012232.01223
Root analytic conductor: 1.418531.41853
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ252(169,⋅)\chi_{252} (169, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 252, ( :1/2), 0.998+0.0576i)(2,\ 252,\ (\ :1/2),\ 0.998 + 0.0576i)

Particular Values

L(1)L(1) ≈\approx 1.74933−0.0504494i1.74933 - 0.0504494i
L(12)L(\frac12) ≈\approx 1.74933−0.0504494i1.74933 - 0.0504494i
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
3 1+(−1.64+0.545i)T 1 + (-1.64 + 0.545i)T
7 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good5 1+(−0.849−1.47i)T+(−2.5+4.33i)T2 1 + (-0.849 - 1.47i)T + (-2.5 + 4.33i)T^{2}
11 1+(1.23−2.14i)T+(−5.5−9.52i)T2 1 + (1.23 - 2.14i)T + (-5.5 - 9.52i)T^{2}
13 1+(0.388+0.673i)T+(−6.5+11.2i)T2 1 + (0.388 + 0.673i)T + (-6.5 + 11.2i)T^{2}
17 1−2.81T+17T2 1 - 2.81T + 17T^{2}
19 1+4.98T+19T2 1 + 4.98T + 19T^{2}
23 1+(0.356+0.616i)T+(−11.5+19.9i)T2 1 + (0.356 + 0.616i)T + (-11.5 + 19.9i)T^{2}
29 1+(2.25−3.90i)T+(−14.5−25.1i)T2 1 + (2.25 - 3.90i)T + (-14.5 - 25.1i)T^{2}
31 1+(2.54+4.41i)T+(−15.5+26.8i)T2 1 + (2.54 + 4.41i)T + (-15.5 + 26.8i)T^{2}
37 1+6.87T+37T2 1 + 6.87T + 37T^{2}
41 1+(−2.93−5.08i)T+(−20.5+35.5i)T2 1 + (-2.93 - 5.08i)T + (-20.5 + 35.5i)T^{2}
43 1+(−2.32+4.03i)T+(−21.5−37.2i)T2 1 + (-2.32 + 4.03i)T + (-21.5 - 37.2i)T^{2}
47 1+(6.49−11.2i)T+(−23.5−40.7i)T2 1 + (6.49 - 11.2i)T + (-23.5 - 40.7i)T^{2}
53 1−1.88T+53T2 1 - 1.88T + 53T^{2}
59 1+(7.14+12.3i)T+(−29.5+51.0i)T2 1 + (7.14 + 12.3i)T + (-29.5 + 51.0i)T^{2}
61 1+(7.15−12.3i)T+(−30.5−52.8i)T2 1 + (7.15 - 12.3i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.99+6.91i)T+(−33.5+58.0i)T2 1 + (3.99 + 6.91i)T + (-33.5 + 58.0i)T^{2}
71 1+10.2T+71T2 1 + 10.2T + 71T^{2}
73 1−4.98T+73T2 1 - 4.98T + 73T^{2}
79 1+(−4.60+7.97i)T+(−39.5−68.4i)T2 1 + (-4.60 + 7.97i)T + (-39.5 - 68.4i)T^{2}
83 1+(4.40−7.63i)T+(−41.5−71.8i)T2 1 + (4.40 - 7.63i)T + (-41.5 - 71.8i)T^{2}
89 1−9.65T+89T2 1 - 9.65T + 89T^{2}
97 1+(4.32−7.48i)T+(−48.5−84.0i)T2 1 + (4.32 - 7.48i)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

−12.32204655448213822019311200786, −10.83032114973578574786466118747, −10.12460296603982613872253832124, −9.155258042267239360438596881160, −7.999846285896156494990053914020, −7.23418128840293447553933237774, −6.19569580088585289446433324955, −4.54728175548612171116546703887, −3.17765900896136957523199323648, −1.94969095509039845999500024896, 1.89681628225916855404210113582, 3.36413129489700293149825330547, 4.71448675895617768107620907995, 5.77916325892318407706627560349, 7.33073805670780839804497214772, 8.452264684880892918012076639256, 8.967544552713325109932539769882, 10.00857783404115280157127025302, 10.92448305492503022204835730889, 12.26050169677894480813023990743

Graph of the ZZ-function along the critical line