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Properties

Label 2-1098-9.4-c1-0-15
Degree 22
Conductor 10981098
Sign 0.376−0.926i0.376 - 0.926i
Analytic cond. 8.767578.76757
Root an. cond. 2.961002.96100
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  + (−0.5 − 0.866i)2-s + (1.20 + 1.24i)3-s + (−0.499 + 0.866i)4-s + (−0.417 + 0.723i)5-s + (0.479 − 1.66i)6-s + (−0.864 − 1.49i)7-s + 0.999·8-s + (−0.110 + 2.99i)9-s + 0.835·10-s + (−0.501 − 0.869i)11-s + (−1.68 + 0.417i)12-s + (−0.749 + 1.29i)13-s + (−0.864 + 1.49i)14-s + (−1.40 + 0.348i)15-s + (−0.5 − 0.866i)16-s + 1.43·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.693 + 0.720i)3-s + (−0.249 + 0.433i)4-s + (−0.186 + 0.323i)5-s + (0.195 − 0.679i)6-s + (−0.326 − 0.565i)7-s + 0.353·8-s + (−0.0369 + 0.999i)9-s + 0.264·10-s + (−0.151 − 0.262i)11-s + (−0.485 + 0.120i)12-s + (−0.207 + 0.360i)13-s + (−0.231 + 0.400i)14-s + (−0.362 + 0.0900i)15-s + (−0.125 − 0.216i)16-s + 0.347·17-s + ⋯

Functional equation

Λ(s)=(1098s/2 ΓC(s) L(s)=((0.376−0.926i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1098s/2 ΓC(s+1/2) L(s)=((0.376−0.926i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10981098    =    2⋅32⋅612 \cdot 3^{2} \cdot 61
Sign: 0.376−0.926i0.376 - 0.926i
Analytic conductor: 8.767578.76757
Root analytic conductor: 2.961002.96100
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1098(733,⋅)\chi_{1098} (733, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1098, ( :1/2), 0.376−0.926i)(2,\ 1098,\ (\ :1/2),\ 0.376 - 0.926i)

Particular Values

L(1)L(1) ≈\approx 1.3560784231.356078423
L(12)L(\frac12) ≈\approx 1.3560784231.356078423
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
3 1+(−1.20−1.24i)T 1 + (-1.20 - 1.24i)T
61 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good5 1+(0.417−0.723i)T+(−2.5−4.33i)T2 1 + (0.417 - 0.723i)T + (-2.5 - 4.33i)T^{2}
7 1+(0.864+1.49i)T+(−3.5+6.06i)T2 1 + (0.864 + 1.49i)T + (-3.5 + 6.06i)T^{2}
11 1+(0.501+0.869i)T+(−5.5+9.52i)T2 1 + (0.501 + 0.869i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.749−1.29i)T+(−6.5−11.2i)T2 1 + (0.749 - 1.29i)T + (-6.5 - 11.2i)T^{2}
17 1−1.43T+17T2 1 - 1.43T + 17T^{2}
19 1−6.31T+19T2 1 - 6.31T + 19T^{2}
23 1+(1.98−3.44i)T+(−11.5−19.9i)T2 1 + (1.98 - 3.44i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.95−5.12i)T+(−14.5+25.1i)T2 1 + (-2.95 - 5.12i)T + (-14.5 + 25.1i)T^{2}
31 1+(3.74−6.49i)T+(−15.5−26.8i)T2 1 + (3.74 - 6.49i)T + (-15.5 - 26.8i)T^{2}
37 1−1.46T+37T2 1 - 1.46T + 37T^{2}
41 1+(2.17−3.77i)T+(−20.5−35.5i)T2 1 + (2.17 - 3.77i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.10+1.91i)T+(−21.5+37.2i)T2 1 + (1.10 + 1.91i)T + (-21.5 + 37.2i)T^{2}
47 1+(−1.60−2.77i)T+(−23.5+40.7i)T2 1 + (-1.60 - 2.77i)T + (-23.5 + 40.7i)T^{2}
53 1+7.59T+53T2 1 + 7.59T + 53T^{2}
59 1+(5.53−9.57i)T+(−29.5−51.0i)T2 1 + (5.53 - 9.57i)T + (-29.5 - 51.0i)T^{2}
67 1+(1.12−1.94i)T+(−33.5−58.0i)T2 1 + (1.12 - 1.94i)T + (-33.5 - 58.0i)T^{2}
71 1+12.2T+71T2 1 + 12.2T + 71T^{2}
73 1−8.86T+73T2 1 - 8.86T + 73T^{2}
79 1+(−3.34−5.79i)T+(−39.5+68.4i)T2 1 + (-3.34 - 5.79i)T + (-39.5 + 68.4i)T^{2}
83 1+(6.98+12.0i)T+(−41.5+71.8i)T2 1 + (6.98 + 12.0i)T + (-41.5 + 71.8i)T^{2}
89 1+0.769T+89T2 1 + 0.769T + 89T^{2}
97 1+(−2.59−4.48i)T+(−48.5+84.0i)T2 1 + (-2.59 - 4.48i)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

−9.991398954078720736921003880215, −9.339830976154369176335334235338, −8.607133344507745464552520207364, −7.58706046576660631039906449815, −7.08920131098779524183609267033, −5.50796472067872352736264223437, −4.56682908242154961096991646109, −3.38939577712861788232868683189, −3.09736719651280003751729595233, −1.49307764228737981360906848148, 0.65193673965809840942270265794, 2.17617278088637581087391258131, 3.25689546362280700260598516869, 4.56649175068457705450726547949, 5.70863999653824271601032516812, 6.42185451994948234095814348700, 7.46428697921868152879914898031, 7.951927988398656018436301316348, 8.735565766596568658892654640657, 9.524544854097308015989169241718

Graph of the ZZ-function along the critical line