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Properties

Label 2-108-9.7-c3-0-0
Degree 22
Conductor 108108
Sign −0.839−0.543i-0.839 - 0.543i
Analytic cond. 6.372206.37220
Root an. cond. 2.524322.52432
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.92 − 11.9i)5-s + (−15.3 + 26.5i)7-s + (−21.9 + 38.0i)11-s + (6.11 + 10.5i)13-s − 76.0·17-s − 44.1·19-s + (−39.3 − 68.0i)23-s + (−33.3 + 57.7i)25-s + (−46.3 + 80.3i)29-s + (71.5 + 123. i)31-s + 425.·35-s − 32.4·37-s + (−167. − 290. i)41-s + (249. − 431. i)43-s + (−140. + 244. i)47-s + ⋯
L(s)  = 1  + (−0.619 − 1.07i)5-s + (−0.829 + 1.43i)7-s + (−0.601 + 1.04i)11-s + (0.130 + 0.226i)13-s − 1.08·17-s − 0.533·19-s + (−0.356 − 0.617i)23-s + (−0.266 + 0.461i)25-s + (−0.297 + 0.514i)29-s + (0.414 + 0.717i)31-s + 2.05·35-s − 0.144·37-s + (−0.639 − 1.10i)41-s + (0.883 − 1.53i)43-s + (−0.437 + 0.757i)47-s + ⋯

Functional equation

Λ(s)=(108s/2 ΓC(s) L(s)=((−0.839−0.543i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(108s/2 ΓC(s+3/2) L(s)=((−0.839−0.543i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 108108    =    22⋅332^{2} \cdot 3^{3}
Sign: −0.839−0.543i-0.839 - 0.543i
Analytic conductor: 6.372206.37220
Root analytic conductor: 2.524322.52432
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ108(73,⋅)\chi_{108} (73, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 108, ( :3/2), −0.839−0.543i)(2,\ 108,\ (\ :3/2),\ -0.839 - 0.543i)

Particular Values

L(2)L(2) ≈\approx 0.101381+0.342747i0.101381 + 0.342747i
L(12)L(\frac12) ≈\approx 0.101381+0.342747i0.101381 + 0.342747i
L(52)L(\frac{5}{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
good5 1+(6.92+11.9i)T+(−62.5+108.i)T2 1 + (6.92 + 11.9i)T + (-62.5 + 108. i)T^{2}
7 1+(15.3−26.5i)T+(−171.5−297.i)T2 1 + (15.3 - 26.5i)T + (-171.5 - 297. i)T^{2}
11 1+(21.9−38.0i)T+(−665.5−1.15e3i)T2 1 + (21.9 - 38.0i)T + (-665.5 - 1.15e3i)T^{2}
13 1+(−6.11−10.5i)T+(−1.09e3+1.90e3i)T2 1 + (-6.11 - 10.5i)T + (-1.09e3 + 1.90e3i)T^{2}
17 1+76.0T+4.91e3T2 1 + 76.0T + 4.91e3T^{2}
19 1+44.1T+6.85e3T2 1 + 44.1T + 6.85e3T^{2}
23 1+(39.3+68.0i)T+(−6.08e3+1.05e4i)T2 1 + (39.3 + 68.0i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(46.3−80.3i)T+(−1.21e4−2.11e4i)T2 1 + (46.3 - 80.3i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+(−71.5−123.i)T+(−1.48e4+2.57e4i)T2 1 + (-71.5 - 123. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+32.4T+5.06e4T2 1 + 32.4T + 5.06e4T^{2}
41 1+(167.+290.i)T+(−3.44e4+5.96e4i)T2 1 + (167. + 290. i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(−249.+431.i)T+(−3.97e4−6.88e4i)T2 1 + (-249. + 431. i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1+(140.−244.i)T+(−5.19e4−8.99e4i)T2 1 + (140. - 244. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1−628.T+1.48e5T2 1 - 628.T + 1.48e5T^{2}
59 1+(−252.−437.i)T+(−1.02e5+1.77e5i)T2 1 + (-252. - 437. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(185.−322.i)T+(−1.13e5−1.96e5i)T2 1 + (185. - 322. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(−81.3−140.i)T+(−1.50e5+2.60e5i)T2 1 + (-81.3 - 140. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+433.T+3.57e5T2 1 + 433.T + 3.57e5T^{2}
73 1+629.T+3.89e5T2 1 + 629.T + 3.89e5T^{2}
79 1+(86.3−149.i)T+(−2.46e5−4.26e5i)T2 1 + (86.3 - 149. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+(87.4−151.i)T+(−2.85e5−4.95e5i)T2 1 + (87.4 - 151. i)T + (-2.85e5 - 4.95e5i)T^{2}
89 1+336.T+7.04e5T2 1 + 336.T + 7.04e5T^{2}
97 1+(42.1−73.0i)T+(−4.56e5−7.90e5i)T2 1 + (42.1 - 73.0i)T + (-4.56e5 - 7.90e5i)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

−13.29670154789053497155666878545, −12.43635733177369012877843130152, −12.01416624165151729026167647548, −10.40518937939229141855552676753, −9.042256222058445192317126145184, −8.550069334183797666421594853323, −6.94385304599567988726099320141, −5.50951495613372034614665174932, −4.31112586165925539767884154710, −2.34826370692623216874775066081, 0.18563660148585931195432112021, 3.04706328400482937573617197524, 4.10051637422683345070103402478, 6.20517444479461668825884209700, 7.14747505049895992718645475207, 8.169109422463150787506551977660, 9.853130978317407017011754922790, 10.78991264197072477671999815237, 11.38700439896862608137036464759, 13.15281951288851388844291446793

Graph of the ZZ-function along the critical line