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Properties

Label 2-18e2-9.2-c8-0-1
Degree 22
Conductor 324324
Sign −0.642+0.766i-0.642 + 0.766i
Analytic cond. 131.990131.990
Root an. cond. 11.488711.4887
Motivic weight 88
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.01e3 + 3.49e3i)7-s + (1.79e4 + 3.10e4i)13-s − 2.58e5·19-s + (−1.95e5 + 3.38e5i)25-s + (9.04e5 + 1.56e6i)31-s + 5.03e5·37-s + (−1.74e6 + 3.02e6i)43-s + (−5.25e6 − 9.10e6i)49-s + (1.19e7 − 2.06e7i)61-s + (2.71e6 + 4.69e6i)67-s + 1.61e7·73-s + (9.44e6 − 1.63e7i)79-s − 1.44e8·91-s + (−8.84e7 + 1.53e8i)97-s + (−2.22e7 − 3.84e7i)103-s + ⋯
L(s)  = 1  + (−0.840 + 1.45i)7-s + (0.626 + 1.08i)13-s − 1.98·19-s + (−0.5 + 0.866i)25-s + (0.979 + 1.69i)31-s + 0.268·37-s + (−0.510 + 0.884i)43-s + (−0.911 − 1.57i)49-s + (0.860 − 1.49i)61-s + (0.134 + 0.232i)67-s + 0.569·73-s + (0.242 − 0.419i)79-s − 2.10·91-s + (−0.999 + 1.73i)97-s + (−0.197 − 0.342i)103-s + ⋯

Functional equation

Λ(s)=(324s/2 ΓC(s) L(s)=((−0.642+0.766i) Λ‾(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(9-s) \end{aligned}
Λ(s)=(324s/2 ΓC(s+4) L(s)=((−0.642+0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 324324    =    22⋅342^{2} \cdot 3^{4}
Sign: −0.642+0.766i-0.642 + 0.766i
Analytic conductor: 131.990131.990
Root analytic conductor: 11.488711.4887
Motivic weight: 88
Rational: no
Arithmetic: yes
Character: χ324(269,⋅)\chi_{324} (269, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 324, ( :4), −0.642+0.766i)(2,\ 324,\ (\ :4),\ -0.642 + 0.766i)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 0.47337806300.4733780630
L(12)L(\frac12) ≈\approx 0.47337806300.4733780630
L(5)L(5) 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
good5 1+(1.95e5−3.38e5i)T2 1 + (1.95e5 - 3.38e5i)T^{2}
7 1+(2.01e3−3.49e3i)T+(−2.88e6−4.99e6i)T2 1 + (2.01e3 - 3.49e3i)T + (-2.88e6 - 4.99e6i)T^{2}
11 1+(1.07e8+1.85e8i)T2 1 + (1.07e8 + 1.85e8i)T^{2}
13 1+(−1.79e4−3.10e4i)T+(−4.07e8+7.06e8i)T2 1 + (-1.79e4 - 3.10e4i)T + (-4.07e8 + 7.06e8i)T^{2}
17 1−6.97e9T2 1 - 6.97e9T^{2}
19 1+2.58e5T+1.69e10T2 1 + 2.58e5T + 1.69e10T^{2}
23 1+(3.91e10−6.78e10i)T2 1 + (3.91e10 - 6.78e10i)T^{2}
29 1+(2.50e11+4.33e11i)T2 1 + (2.50e11 + 4.33e11i)T^{2}
31 1+(−9.04e5−1.56e6i)T+(−4.26e11+7.38e11i)T2 1 + (-9.04e5 - 1.56e6i)T + (-4.26e11 + 7.38e11i)T^{2}
37 1−5.03e5T+3.51e12T2 1 - 5.03e5T + 3.51e12T^{2}
41 1+(3.99e12−6.91e12i)T2 1 + (3.99e12 - 6.91e12i)T^{2}
43 1+(1.74e6−3.02e6i)T+(−5.84e12−1.01e13i)T2 1 + (1.74e6 - 3.02e6i)T + (-5.84e12 - 1.01e13i)T^{2}
47 1+(1.19e13+2.06e13i)T2 1 + (1.19e13 + 2.06e13i)T^{2}
53 1−6.22e13T2 1 - 6.22e13T^{2}
59 1+(7.34e13−1.27e14i)T2 1 + (7.34e13 - 1.27e14i)T^{2}
61 1+(−1.19e7+2.06e7i)T+(−9.58e13−1.66e14i)T2 1 + (-1.19e7 + 2.06e7i)T + (-9.58e13 - 1.66e14i)T^{2}
67 1+(−2.71e6−4.69e6i)T+(−2.03e14+3.51e14i)T2 1 + (-2.71e6 - 4.69e6i)T + (-2.03e14 + 3.51e14i)T^{2}
71 1−6.45e14T2 1 - 6.45e14T^{2}
73 1−1.61e7T+8.06e14T2 1 - 1.61e7T + 8.06e14T^{2}
79 1+(−9.44e6+1.63e7i)T+(−7.58e14−1.31e15i)T2 1 + (-9.44e6 + 1.63e7i)T + (-7.58e14 - 1.31e15i)T^{2}
83 1+(1.12e15+1.95e15i)T2 1 + (1.12e15 + 1.95e15i)T^{2}
89 1−3.93e15T2 1 - 3.93e15T^{2}
97 1+(8.84e7−1.53e8i)T+(−3.91e15−6.78e15i)T2 1 + (8.84e7 - 1.53e8i)T + (-3.91e15 - 6.78e15i)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

−10.87659784576660504178985328767, −9.723228351268854373709354098227, −8.925777560684071260822647622830, −8.306940146030782519466177975796, −6.65203404376633550482421383891, −6.22715716114645624361080937818, −5.01241724588277672009945684442, −3.77105350827286726438428774383, −2.61724699338215760435855471286, −1.64287966455978582990305820425, 0.11582544503056237028526641737, 0.837357573385050783306262478861, 2.41460745194497708422375330832, 3.70005543371782148132541209702, 4.36156935697647920845673272182, 5.95636965011914742950809370120, 6.65822614359063744180628316725, 7.75469643550362731630167330637, 8.572922088094647786347454738559, 9.970639523467609981163765214133

Graph of the ZZ-function along the critical line