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Properties

Label 2-30e2-100.23-c1-0-36
Degree 22
Conductor 900900
Sign 0.817−0.576i0.817 - 0.576i
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  + (−0.618 + 1.27i)2-s + (−1.23 − 1.57i)4-s + (2.17 − 0.500i)5-s + (−0.974 − 0.974i)7-s + (2.76 − 0.595i)8-s + (−0.711 + 3.08i)10-s + (1.28 + 1.77i)11-s + (0.471 + 2.97i)13-s + (1.84 − 0.636i)14-s + (−0.953 + 3.88i)16-s + (0.167 + 0.0855i)17-s + (−2.61 − 8.06i)19-s + (−3.47 − 2.81i)20-s + (−3.04 + 0.540i)22-s + (−0.127 + 0.802i)23-s + ⋯
L(s)  = 1  + (−0.437 + 0.899i)2-s + (−0.617 − 0.786i)4-s + (0.974 − 0.224i)5-s + (−0.368 − 0.368i)7-s + (0.977 − 0.210i)8-s + (−0.224 + 0.974i)10-s + (0.387 + 0.533i)11-s + (0.130 + 0.826i)13-s + (0.492 − 0.170i)14-s + (−0.238 + 0.971i)16-s + (0.0407 + 0.0207i)17-s + (−0.600 − 1.84i)19-s + (−0.777 − 0.628i)20-s + (−0.649 + 0.115i)22-s + (−0.0264 + 0.167i)23-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((0.817−0.576i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s+1/2) L(s)=((0.817−0.576i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: 0.817−0.576i0.817 - 0.576i
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), 0.817−0.576i)(2,\ 900,\ (\ :1/2),\ 0.817 - 0.576i)

Particular Values

L(1)L(1) ≈\approx 1.34155+0.425567i1.34155 + 0.425567i
L(12)L(\frac12) ≈\approx 1.34155+0.425567i1.34155 + 0.425567i
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+(0.618−1.27i)T 1 + (0.618 - 1.27i)T
3 1 1
5 1+(−2.17+0.500i)T 1 + (-2.17 + 0.500i)T
good7 1+(0.974+0.974i)T+7iT2 1 + (0.974 + 0.974i)T + 7iT^{2}
11 1+(−1.28−1.77i)T+(−3.39+10.4i)T2 1 + (-1.28 - 1.77i)T + (-3.39 + 10.4i)T^{2}
13 1+(−0.471−2.97i)T+(−12.3+4.01i)T2 1 + (-0.471 - 2.97i)T + (-12.3 + 4.01i)T^{2}
17 1+(−0.167−0.0855i)T+(9.99+13.7i)T2 1 + (-0.167 - 0.0855i)T + (9.99 + 13.7i)T^{2}
19 1+(2.61+8.06i)T+(−15.3+11.1i)T2 1 + (2.61 + 8.06i)T + (-15.3 + 11.1i)T^{2}
23 1+(0.127−0.802i)T+(−21.8−7.10i)T2 1 + (0.127 - 0.802i)T + (-21.8 - 7.10i)T^{2}
29 1+(−9.37−3.04i)T+(23.4+17.0i)T2 1 + (-9.37 - 3.04i)T + (23.4 + 17.0i)T^{2}
31 1+(−4.18+1.35i)T+(25.0−18.2i)T2 1 + (-4.18 + 1.35i)T + (25.0 - 18.2i)T^{2}
37 1+(−6.96+1.10i)T+(35.1−11.4i)T2 1 + (-6.96 + 1.10i)T + (35.1 - 11.4i)T^{2}
41 1+(−2.94−2.13i)T+(12.6+38.9i)T2 1 + (-2.94 - 2.13i)T + (12.6 + 38.9i)T^{2}
43 1+(−1.86+1.86i)T−43iT2 1 + (-1.86 + 1.86i)T - 43iT^{2}
47 1+(−5.26+2.68i)T+(27.6−38.0i)T2 1 + (-5.26 + 2.68i)T + (27.6 - 38.0i)T^{2}
53 1+(6.90−3.51i)T+(31.1−42.8i)T2 1 + (6.90 - 3.51i)T + (31.1 - 42.8i)T^{2}
59 1+(−4.74−3.44i)T+(18.2+56.1i)T2 1 + (-4.74 - 3.44i)T + (18.2 + 56.1i)T^{2}
61 1+(2.78−2.02i)T+(18.8−58.0i)T2 1 + (2.78 - 2.02i)T + (18.8 - 58.0i)T^{2}
67 1+(3.24−6.37i)T+(−39.3−54.2i)T2 1 + (3.24 - 6.37i)T + (-39.3 - 54.2i)T^{2}
71 1+(9.69+3.15i)T+(57.4+41.7i)T2 1 + (9.69 + 3.15i)T + (57.4 + 41.7i)T^{2}
73 1+(−0.675−0.106i)T+(69.4+22.5i)T2 1 + (-0.675 - 0.106i)T + (69.4 + 22.5i)T^{2}
79 1+(−3.74+11.5i)T+(−63.9−46.4i)T2 1 + (-3.74 + 11.5i)T + (-63.9 - 46.4i)T^{2}
83 1+(−6.86−3.49i)T+(48.7+67.1i)T2 1 + (-6.86 - 3.49i)T + (48.7 + 67.1i)T^{2}
89 1+(4.99+6.87i)T+(−27.5+84.6i)T2 1 + (4.99 + 6.87i)T + (-27.5 + 84.6i)T^{2}
97 1+(−5.10−10.0i)T+(−57.0+78.4i)T2 1 + (-5.10 - 10.0i)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

−10.01320456449300337595669146425, −9.113419127299011426350001881606, −8.819168191176694236801380270392, −7.49217050708243679998259413075, −6.62170077008818817032201345384, −6.24657707546452886033537831536, −4.93200963397160234783313608672, −4.33132277298159845095498766074, −2.45714919107679564474897396889, −1.02390938049596579422891136698, 1.14076735312036848559820095663, 2.45759036372337649601556954618, 3.27913860148971590669263668128, 4.49020916029612541823527517238, 5.80520819154123507992458665980, 6.39353468927249513802973373755, 7.85817919070051953085948654494, 8.500630821460245282215968381391, 9.407345980198549673744160232390, 10.12979928297385538365501922851

Graph of the ZZ-function along the critical line