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Properties

Label 2-10e3-25.4-c1-0-21
Degree 22
Conductor 10001000
Sign −0.628+0.777i-0.628 + 0.777i
Analytic cond. 7.985047.98504
Root an. cond. 2.825782.82578
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  + (1.98 − 2.72i)3-s − 0.794i·7-s + (−2.58 − 7.94i)9-s + (0.673 − 2.07i)11-s + (2.73 − 0.888i)13-s + (−0.0736 − 0.101i)17-s + (−5.66 + 4.11i)19-s + (−2.16 − 1.57i)21-s + (5.38 + 1.74i)23-s + (−17.1 − 5.57i)27-s + (0.989 + 0.718i)29-s + (−5.53 + 4.01i)31-s + (−4.31 − 5.94i)33-s + (6.55 − 2.12i)37-s + (2.99 − 9.21i)39-s + ⋯
L(s)  = 1  + (1.14 − 1.57i)3-s − 0.300i·7-s + (−0.860 − 2.64i)9-s + (0.203 − 0.624i)11-s + (0.758 − 0.246i)13-s + (−0.0178 − 0.0245i)17-s + (−1.29 + 0.943i)19-s + (−0.472 − 0.343i)21-s + (1.12 + 0.364i)23-s + (−3.30 − 1.07i)27-s + (0.183 + 0.133i)29-s + (−0.993 + 0.721i)31-s + (−0.751 − 1.03i)33-s + (1.07 − 0.350i)37-s + (0.479 − 1.47i)39-s + ⋯

Functional equation

Λ(s)=(1000s/2 ΓC(s) L(s)=((−0.628+0.777i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1000s/2 ΓC(s+1/2) L(s)=((−0.628+0.777i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10001000    =    23⋅532^{3} \cdot 5^{3}
Sign: −0.628+0.777i-0.628 + 0.777i
Analytic conductor: 7.985047.98504
Root analytic conductor: 2.825782.82578
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1000(649,⋅)\chi_{1000} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1000, ( :1/2), −0.628+0.777i)(2,\ 1000,\ (\ :1/2),\ -0.628 + 0.777i)

Particular Values

L(1)L(1) ≈\approx 0.970304−2.03260i0.970304 - 2.03260i
L(12)L(\frac12) ≈\approx 0.970304−2.03260i0.970304 - 2.03260i
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
5 1 1
good3 1+(−1.98+2.72i)T+(−0.927−2.85i)T2 1 + (-1.98 + 2.72i)T + (-0.927 - 2.85i)T^{2}
7 1+0.794iT−7T2 1 + 0.794iT - 7T^{2}
11 1+(−0.673+2.07i)T+(−8.89−6.46i)T2 1 + (-0.673 + 2.07i)T + (-8.89 - 6.46i)T^{2}
13 1+(−2.73+0.888i)T+(10.5−7.64i)T2 1 + (-2.73 + 0.888i)T + (10.5 - 7.64i)T^{2}
17 1+(0.0736+0.101i)T+(−5.25+16.1i)T2 1 + (0.0736 + 0.101i)T + (-5.25 + 16.1i)T^{2}
19 1+(5.66−4.11i)T+(5.87−18.0i)T2 1 + (5.66 - 4.11i)T + (5.87 - 18.0i)T^{2}
23 1+(−5.38−1.74i)T+(18.6+13.5i)T2 1 + (-5.38 - 1.74i)T + (18.6 + 13.5i)T^{2}
29 1+(−0.989−0.718i)T+(8.96+27.5i)T2 1 + (-0.989 - 0.718i)T + (8.96 + 27.5i)T^{2}
31 1+(5.53−4.01i)T+(9.57−29.4i)T2 1 + (5.53 - 4.01i)T + (9.57 - 29.4i)T^{2}
37 1+(−6.55+2.12i)T+(29.9−21.7i)T2 1 + (-6.55 + 2.12i)T + (29.9 - 21.7i)T^{2}
41 1+(−0.244−0.753i)T+(−33.1+24.0i)T2 1 + (-0.244 - 0.753i)T + (-33.1 + 24.0i)T^{2}
43 1+6.69iT−43T2 1 + 6.69iT - 43T^{2}
47 1+(1.79−2.46i)T+(−14.5−44.6i)T2 1 + (1.79 - 2.46i)T + (-14.5 - 44.6i)T^{2}
53 1+(−0.536+0.738i)T+(−16.3−50.4i)T2 1 + (-0.536 + 0.738i)T + (-16.3 - 50.4i)T^{2}
59 1+(−0.853−2.62i)T+(−47.7+34.6i)T2 1 + (-0.853 - 2.62i)T + (-47.7 + 34.6i)T^{2}
61 1+(−3.22+9.92i)T+(−49.3−35.8i)T2 1 + (-3.22 + 9.92i)T + (-49.3 - 35.8i)T^{2}
67 1+(−2.90−3.99i)T+(−20.7+63.7i)T2 1 + (-2.90 - 3.99i)T + (-20.7 + 63.7i)T^{2}
71 1+(6.39+4.64i)T+(21.9+67.5i)T2 1 + (6.39 + 4.64i)T + (21.9 + 67.5i)T^{2}
73 1+(3.01+0.980i)T+(59.0+42.9i)T2 1 + (3.01 + 0.980i)T + (59.0 + 42.9i)T^{2}
79 1+(−1.78−1.29i)T+(24.4+75.1i)T2 1 + (-1.78 - 1.29i)T + (24.4 + 75.1i)T^{2}
83 1+(−0.280−0.385i)T+(−25.6+78.9i)T2 1 + (-0.280 - 0.385i)T + (-25.6 + 78.9i)T^{2}
89 1+(4.61−14.2i)T+(−72.0−52.3i)T2 1 + (4.61 - 14.2i)T + (-72.0 - 52.3i)T^{2}
97 1+(−5.06+6.97i)T+(−29.9−92.2i)T2 1 + (-5.06 + 6.97i)T + (-29.9 - 92.2i)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

−9.262482663483781438846877059381, −8.662606087830128379651997720355, −8.060223393021155012837582519229, −7.22657489364056970653449198944, −6.49525904130617708140136815217, −5.71964732655714333096403700504, −3.90477571876458279540926014913, −3.16056885635640287419348540109, −1.98607454804731189775819142244, −0.932657599038048472474150248139, 2.16373691864068546523066903074, 3.05325462393104379251824679035, 4.18298905347806903505279946108, 4.62649369372463264886361141964, 5.77652475044411611596766203036, 7.03943212827518439710998206203, 8.151910946723141702748591509564, 8.858287592140459419388498602940, 9.287440349047988971501157610534, 10.13693437938864532862904731015

Graph of the ZZ-function along the critical line