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Properties

Label 2-105-105.2-c1-0-10
Degree 22
Conductor 105105
Sign −0.710+0.704i-0.710 + 0.704i
Analytic cond. 0.8384290.838429
Root an. cond. 0.9156570.915657
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  + (−0.391 − 1.46i)2-s + (−0.879 − 1.49i)3-s + (−0.246 + 0.142i)4-s + (1.82 − 1.29i)5-s + (−1.83 + 1.86i)6-s + (−1.17 + 2.36i)7-s + (−1.83 − 1.83i)8-s + (−1.45 + 2.62i)9-s + (−2.60 − 2.15i)10-s + (−0.791 + 0.457i)11-s + (0.429 + 0.243i)12-s + (3.07 − 3.07i)13-s + (3.92 + 0.791i)14-s + (−3.53 − 1.58i)15-s + (−2.24 + 3.88i)16-s + (1.16 + 0.311i)17-s + ⋯
L(s)  = 1  + (−0.276 − 1.03i)2-s + (−0.507 − 0.861i)3-s + (−0.123 + 0.0712i)4-s + (0.815 − 0.578i)5-s + (−0.749 + 0.762i)6-s + (−0.444 + 0.895i)7-s + (−0.648 − 0.648i)8-s + (−0.484 + 0.874i)9-s + (−0.822 − 0.682i)10-s + (−0.238 + 0.137i)11-s + (0.124 + 0.0702i)12-s + (0.854 − 0.854i)13-s + (1.04 + 0.211i)14-s + (−0.912 − 0.409i)15-s + (−0.561 + 0.971i)16-s + (0.281 + 0.0755i)17-s + ⋯

Functional equation

Λ(s)=(105s/2 ΓC(s) L(s)=((−0.710+0.704i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(105s/2 ΓC(s+1/2) L(s)=((−0.710+0.704i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 105105    =    3⋅5⋅73 \cdot 5 \cdot 7
Sign: −0.710+0.704i-0.710 + 0.704i
Analytic conductor: 0.8384290.838429
Root analytic conductor: 0.9156570.915657
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ105(2,⋅)\chi_{105} (2, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 105, ( :1/2), −0.710+0.704i)(2,\ 105,\ (\ :1/2),\ -0.710 + 0.704i)

Particular Values

L(1)L(1) ≈\approx 0.331298−0.804514i0.331298 - 0.804514i
L(12)L(\frac12) ≈\approx 0.331298−0.804514i0.331298 - 0.804514i
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)
bad3 1+(0.879+1.49i)T 1 + (0.879 + 1.49i)T
5 1+(−1.82+1.29i)T 1 + (-1.82 + 1.29i)T
7 1+(1.17−2.36i)T 1 + (1.17 - 2.36i)T
good2 1+(0.391+1.46i)T+(−1.73+i)T2 1 + (0.391 + 1.46i)T + (-1.73 + i)T^{2}
11 1+(0.791−0.457i)T+(5.5−9.52i)T2 1 + (0.791 - 0.457i)T + (5.5 - 9.52i)T^{2}
13 1+(−3.07+3.07i)T−13iT2 1 + (-3.07 + 3.07i)T - 13iT^{2}
17 1+(−1.16−0.311i)T+(14.7+8.5i)T2 1 + (-1.16 - 0.311i)T + (14.7 + 8.5i)T^{2}
19 1+(−5.95−3.43i)T+(9.5+16.4i)T2 1 + (-5.95 - 3.43i)T + (9.5 + 16.4i)T^{2}
23 1+(1.88−0.505i)T+(19.9−11.5i)T2 1 + (1.88 - 0.505i)T + (19.9 - 11.5i)T^{2}
29 1−2.72T+29T2 1 - 2.72T + 29T^{2}
31 1+(2.31+4.01i)T+(−15.5+26.8i)T2 1 + (2.31 + 4.01i)T + (-15.5 + 26.8i)T^{2}
37 1+(0.774−0.207i)T+(32.0−18.5i)T2 1 + (0.774 - 0.207i)T + (32.0 - 18.5i)T^{2}
41 1−0.922iT−41T2 1 - 0.922iT - 41T^{2}
43 1+(4.80−4.80i)T−43iT2 1 + (4.80 - 4.80i)T - 43iT^{2}
47 1+(−2.71−10.1i)T+(−40.7+23.5i)T2 1 + (-2.71 - 10.1i)T + (-40.7 + 23.5i)T^{2}
53 1+(2.85−10.6i)T+(−45.8−26.5i)T2 1 + (2.85 - 10.6i)T + (-45.8 - 26.5i)T^{2}
59 1+(4.94+8.55i)T+(−29.5+51.0i)T2 1 + (4.94 + 8.55i)T + (-29.5 + 51.0i)T^{2}
61 1+(−0.533+0.924i)T+(−30.5−52.8i)T2 1 + (-0.533 + 0.924i)T + (-30.5 - 52.8i)T^{2}
67 1+(1.83−6.83i)T+(−58.0−33.5i)T2 1 + (1.83 - 6.83i)T + (-58.0 - 33.5i)T^{2}
71 1−0.557iT−71T2 1 - 0.557iT - 71T^{2}
73 1+(−2.10−0.564i)T+(63.2+36.5i)T2 1 + (-2.10 - 0.564i)T + (63.2 + 36.5i)T^{2}
79 1+(−2.62−1.51i)T+(39.5+68.4i)T2 1 + (-2.62 - 1.51i)T + (39.5 + 68.4i)T^{2}
83 1+(2.38+2.38i)T+83iT2 1 + (2.38 + 2.38i)T + 83iT^{2}
89 1+(5.64−9.78i)T+(−44.5−77.0i)T2 1 + (5.64 - 9.78i)T + (-44.5 - 77.0i)T^{2}
97 1+(1.58+1.58i)T+97iT2 1 + (1.58 + 1.58i)T + 97iT^{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

−12.86768900673364318847468579194, −12.39553590504107012548461111668, −11.42643746516770855696824855715, −10.26746456174484044721470634627, −9.344510785224519598349934695339, −8.038144938338322050800212712427, −6.22711759228484448306377298645, −5.55437632177330768590485965859, −2.87084453831115097566146032049, −1.38870816652073950795454864894, 3.36232665685655901241319687431, 5.25421233589555496593630846364, 6.38046539980900763529485975758, 7.14220743513215475832518816371, 8.833165568335685518595688298254, 9.828015966412588325421936373741, 10.81741786145503840120653668215, 11.79821094049667137042577187372, 13.61571878366485080399089341272, 14.27465968985452144741059649516

Graph of the ZZ-function along the critical line