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Properties

Label 2-30e2-100.23-c1-0-18
Degree 22
Conductor 900900
Sign 0.299−0.954i0.299 - 0.954i
Analytic cond. 7.186537.18653
Root an. cond. 2.680772.68077
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  + (−1.24 + 0.678i)2-s + (1.08 − 1.68i)4-s + (0.872 − 2.05i)5-s + (3.13 + 3.13i)7-s + (−0.199 + 2.82i)8-s + (0.312 + 3.14i)10-s + (3.28 + 4.52i)11-s + (0.499 + 3.15i)13-s + (−6.00 − 1.76i)14-s + (−1.66 − 3.63i)16-s + (−2.24 − 1.14i)17-s + (−0.732 − 2.25i)19-s + (−2.52 − 3.69i)20-s + (−7.14 − 3.38i)22-s + (−1.33 + 8.40i)23-s + ⋯
L(s)  = 1  + (−0.877 + 0.479i)2-s + (0.540 − 0.841i)4-s + (0.390 − 0.920i)5-s + (1.18 + 1.18i)7-s + (−0.0704 + 0.997i)8-s + (0.0989 + 0.995i)10-s + (0.990 + 1.36i)11-s + (0.138 + 0.875i)13-s + (−1.60 − 0.471i)14-s + (−0.416 − 0.909i)16-s + (−0.543 − 0.276i)17-s + (−0.167 − 0.516i)19-s + (−0.563 − 0.825i)20-s + (−1.52 − 0.721i)22-s + (−0.277 + 1.75i)23-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((0.299−0.954i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s+1/2) L(s)=((0.299−0.954i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: 0.299−0.954i0.299 - 0.954i
Analytic conductor: 7.186537.18653
Root analytic conductor: 2.680772.68077
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ900(523,⋅)\chi_{900} (523, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :1/2), 0.299−0.954i)(2,\ 900,\ (\ :1/2),\ 0.299 - 0.954i)

Particular Values

L(1)L(1) ≈\approx 1.01586+0.745606i1.01586 + 0.745606i
L(12)L(\frac12) ≈\approx 1.01586+0.745606i1.01586 + 0.745606i
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+(1.24−0.678i)T 1 + (1.24 - 0.678i)T
3 1 1
5 1+(−0.872+2.05i)T 1 + (-0.872 + 2.05i)T
good7 1+(−3.13−3.13i)T+7iT2 1 + (-3.13 - 3.13i)T + 7iT^{2}
11 1+(−3.28−4.52i)T+(−3.39+10.4i)T2 1 + (-3.28 - 4.52i)T + (-3.39 + 10.4i)T^{2}
13 1+(−0.499−3.15i)T+(−12.3+4.01i)T2 1 + (-0.499 - 3.15i)T + (-12.3 + 4.01i)T^{2}
17 1+(2.24+1.14i)T+(9.99+13.7i)T2 1 + (2.24 + 1.14i)T + (9.99 + 13.7i)T^{2}
19 1+(0.732+2.25i)T+(−15.3+11.1i)T2 1 + (0.732 + 2.25i)T + (-15.3 + 11.1i)T^{2}
23 1+(1.33−8.40i)T+(−21.8−7.10i)T2 1 + (1.33 - 8.40i)T + (-21.8 - 7.10i)T^{2}
29 1+(3.85+1.25i)T+(23.4+17.0i)T2 1 + (3.85 + 1.25i)T + (23.4 + 17.0i)T^{2}
31 1+(2.06−0.670i)T+(25.0−18.2i)T2 1 + (2.06 - 0.670i)T + (25.0 - 18.2i)T^{2}
37 1+(3.19−0.506i)T+(35.1−11.4i)T2 1 + (3.19 - 0.506i)T + (35.1 - 11.4i)T^{2}
41 1+(−5.37−3.90i)T+(12.6+38.9i)T2 1 + (-5.37 - 3.90i)T + (12.6 + 38.9i)T^{2}
43 1+(1.00−1.00i)T−43iT2 1 + (1.00 - 1.00i)T - 43iT^{2}
47 1+(−0.610+0.311i)T+(27.6−38.0i)T2 1 + (-0.610 + 0.311i)T + (27.6 - 38.0i)T^{2}
53 1+(−8.35+4.25i)T+(31.1−42.8i)T2 1 + (-8.35 + 4.25i)T + (31.1 - 42.8i)T^{2}
59 1+(3.98+2.89i)T+(18.2+56.1i)T2 1 + (3.98 + 2.89i)T + (18.2 + 56.1i)T^{2}
61 1+(2.67−1.94i)T+(18.8−58.0i)T2 1 + (2.67 - 1.94i)T + (18.8 - 58.0i)T^{2}
67 1+(−0.883+1.73i)T+(−39.3−54.2i)T2 1 + (-0.883 + 1.73i)T + (-39.3 - 54.2i)T^{2}
71 1+(7.28+2.36i)T+(57.4+41.7i)T2 1 + (7.28 + 2.36i)T + (57.4 + 41.7i)T^{2}
73 1+(−7.69−1.21i)T+(69.4+22.5i)T2 1 + (-7.69 - 1.21i)T + (69.4 + 22.5i)T^{2}
79 1+(−1.42+4.39i)T+(−63.9−46.4i)T2 1 + (-1.42 + 4.39i)T + (-63.9 - 46.4i)T^{2}
83 1+(−11.1−5.69i)T+(48.7+67.1i)T2 1 + (-11.1 - 5.69i)T + (48.7 + 67.1i)T^{2}
89 1+(7.91+10.8i)T+(−27.5+84.6i)T2 1 + (7.91 + 10.8i)T + (-27.5 + 84.6i)T^{2}
97 1+(−0.0175−0.0345i)T+(−57.0+78.4i)T2 1 + (-0.0175 - 0.0345i)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

−9.812882883574206740194034778687, −9.187256695268217082685917110904, −8.884644579769301104838150304395, −7.86793698181111876660685654690, −7.02156535473182140451937979781, −5.96666108045230119173607971021, −5.13712175240189413257142728280, −4.39817193663967386376530464927, −2.02427638744406235667860743688, −1.63404243251017512634921796712, 0.851593557924656242711610309833, 2.10264575042650625675910475127, 3.42583636379127914838459896227, 4.16921068162905114592101231191, 5.86151872960368307648299375472, 6.71702417545868882151977952286, 7.56496138515209457959836728966, 8.323685263174783947795957723157, 9.041650609354446061838310029326, 10.29868310980985941832022691546

Graph of the ZZ-function along the critical line