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Properties

Label 2-600-600.77-c1-0-33
Degree 22
Conductor 600600
Sign 0.647−0.762i0.647 - 0.762i
Analytic cond. 4.791024.79102
Root an. cond. 2.188842.18884
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  + (1.39 + 0.221i)2-s + (−0.786 − 1.54i)3-s + (1.90 + 0.618i)4-s + (−1.93 + 1.12i)5-s + (−0.756 − 2.32i)6-s + (3.72 + 3.72i)7-s + (2.52 + 1.28i)8-s + (−1.76 + 2.42i)9-s + (−2.94 + 1.14i)10-s + (−3.48 + 2.53i)11-s + (−0.541 − 3.42i)12-s + (4.38 + 6.03i)14-s + (3.25 + 2.09i)15-s + (3.23 + 2.35i)16-s + (−2.99 + 3i)18-s + ⋯
L(s)  = 1  + (0.987 + 0.156i)2-s + (−0.453 − 0.891i)3-s + (0.951 + 0.309i)4-s + (−0.863 + 0.503i)5-s + (−0.309 − 0.951i)6-s + (1.40 + 1.40i)7-s + (0.891 + 0.453i)8-s + (−0.587 + 0.809i)9-s + (−0.932 + 0.362i)10-s + (−1.05 + 0.763i)11-s + (−0.156 − 0.987i)12-s + (1.17 + 1.61i)14-s + (0.840 + 0.541i)15-s + (0.809 + 0.587i)16-s + (−0.707 + 0.707i)18-s + ⋯

Functional equation

Λ(s)=(600s/2 ΓC(s) L(s)=((0.647−0.762i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(600s/2 ΓC(s+1/2) L(s)=((0.647−0.762i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 600600    =    23⋅3⋅522^{3} \cdot 3 \cdot 5^{2}
Sign: 0.647−0.762i0.647 - 0.762i
Analytic conductor: 4.791024.79102
Root analytic conductor: 2.188842.18884
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ600(77,⋅)\chi_{600} (77, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 600, ( :1/2), 0.647−0.762i)(2,\ 600,\ (\ :1/2),\ 0.647 - 0.762i)

Particular Values

L(1)L(1) ≈\approx 1.97375+0.913321i1.97375 + 0.913321i
L(12)L(\frac12) ≈\approx 1.97375+0.913321i1.97375 + 0.913321i
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+(−1.39−0.221i)T 1 + (-1.39 - 0.221i)T
3 1+(0.786+1.54i)T 1 + (0.786 + 1.54i)T
5 1+(1.93−1.12i)T 1 + (1.93 - 1.12i)T
good7 1+(−3.72−3.72i)T+7iT2 1 + (-3.72 - 3.72i)T + 7iT^{2}
11 1+(3.48−2.53i)T+(3.39−10.4i)T2 1 + (3.48 - 2.53i)T + (3.39 - 10.4i)T^{2}
13 1+(−12.3+4.01i)T2 1 + (-12.3 + 4.01i)T^{2}
17 1+(9.99+13.7i)T2 1 + (9.99 + 13.7i)T^{2}
19 1+(−15.3+11.1i)T2 1 + (-15.3 + 11.1i)T^{2}
23 1+(21.8+7.10i)T2 1 + (21.8 + 7.10i)T^{2}
29 1+(−8.89−2.88i)T+(23.4+17.0i)T2 1 + (-8.89 - 2.88i)T + (23.4 + 17.0i)T^{2}
31 1+(3.40+10.4i)T+(−25.0+18.2i)T2 1 + (3.40 + 10.4i)T + (-25.0 + 18.2i)T^{2}
37 1+(35.1−11.4i)T2 1 + (35.1 - 11.4i)T^{2}
41 1+(−12.6−38.9i)T2 1 + (-12.6 - 38.9i)T^{2}
43 1+43iT2 1 + 43iT^{2}
47 1+(−27.6+38.0i)T2 1 + (-27.6 + 38.0i)T^{2}
53 1+(2.50+4.90i)T+(−31.1+42.8i)T2 1 + (2.50 + 4.90i)T + (-31.1 + 42.8i)T^{2}
59 1+(−5.61+7.72i)T+(−18.2−56.1i)T2 1 + (-5.61 + 7.72i)T + (-18.2 - 56.1i)T^{2}
61 1+(−18.8+58.0i)T2 1 + (-18.8 + 58.0i)T^{2}
67 1+(39.3+54.2i)T2 1 + (39.3 + 54.2i)T^{2}
71 1+(57.4+41.7i)T2 1 + (57.4 + 41.7i)T^{2}
73 1+(0.464−2.93i)T+(−69.4−22.5i)T2 1 + (0.464 - 2.93i)T + (-69.4 - 22.5i)T^{2}
79 1+(5.71+1.85i)T+(63.9+46.4i)T2 1 + (5.71 + 1.85i)T + (63.9 + 46.4i)T^{2}
83 1+(13.1+6.68i)T+(48.7+67.1i)T2 1 + (13.1 + 6.68i)T + (48.7 + 67.1i)T^{2}
89 1+(27.5−84.6i)T2 1 + (27.5 - 84.6i)T^{2}
97 1+(−6.33+3.22i)T+(57.0−78.4i)T2 1 + (-6.33 + 3.22i)T + (57.0 - 78.4i)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

−11.36099336465699744959585035645, −10.39516706084236949888702895502, −8.461609023712259972448505168626, −7.900480190320614053381661806664, −7.23510287636725092999554578765, −6.12724537088375012160055753935, −5.22805457061458657094229977902, −4.55821375408455091607584557789, −2.77183182461067812462264464172, −2.00817711037169617720739150490, 0.995648418896374232789093690326, 3.17419638630023707731814996253, 4.18087059273535018442586525706, 4.78179625251166293800024412009, 5.45637378597326534146631625530, 6.89550306795443290327417643838, 7.84970033893349963681662023553, 8.592330132285158646472188292981, 10.34360044970216043700425220333, 10.63853370142658048873414692725

Graph of the ZZ-function along the critical line