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Properties

Label 2-15e2-225.106-c1-0-5
Degree 22
Conductor 225225
Sign 0.881−0.472i0.881 - 0.472i
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
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  + (−2.47 − 0.526i)2-s + (−1.29 + 1.15i)3-s + (4.03 + 1.79i)4-s + (1.67 − 1.48i)5-s + (3.81 − 2.16i)6-s + (−1.03 + 1.79i)7-s + (−4.94 − 3.59i)8-s + (0.350 − 2.97i)9-s + (−4.92 + 2.80i)10-s + (−2.39 − 0.509i)11-s + (−7.28 + 2.31i)12-s + (1.74 − 0.371i)13-s + (3.50 − 3.89i)14-s + (−0.452 + 3.84i)15-s + (4.44 + 4.94i)16-s + (3.67 + 2.66i)17-s + ⋯
L(s)  = 1  + (−1.75 − 0.372i)2-s + (−0.747 + 0.664i)3-s + (2.01 + 0.897i)4-s + (0.747 − 0.664i)5-s + (1.55 − 0.885i)6-s + (−0.390 + 0.676i)7-s + (−1.74 − 1.26i)8-s + (0.116 − 0.993i)9-s + (−1.55 + 0.885i)10-s + (−0.722 − 0.153i)11-s + (−2.10 + 0.668i)12-s + (0.484 − 0.102i)13-s + (0.936 − 1.04i)14-s + (−0.116 + 0.993i)15-s + (1.11 + 1.23i)16-s + (0.891 + 0.647i)17-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((0.881−0.472i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((0.881−0.472i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: 0.881−0.472i0.881 - 0.472i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ225(106,⋅)\chi_{225} (106, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 225, ( :1/2), 0.881−0.472i)(2,\ 225,\ (\ :1/2),\ 0.881 - 0.472i)

Particular Values

L(1)L(1) ≈\approx 0.456025+0.114427i0.456025 + 0.114427i
L(12)L(\frac12) ≈\approx 0.456025+0.114427i0.456025 + 0.114427i
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)
bad3 1+(1.29−1.15i)T 1 + (1.29 - 1.15i)T
5 1+(−1.67+1.48i)T 1 + (-1.67 + 1.48i)T
good2 1+(2.47+0.526i)T+(1.82+0.813i)T2 1 + (2.47 + 0.526i)T + (1.82 + 0.813i)T^{2}
7 1+(1.03−1.79i)T+(−3.5−6.06i)T2 1 + (1.03 - 1.79i)T + (-3.5 - 6.06i)T^{2}
11 1+(2.39+0.509i)T+(10.0+4.47i)T2 1 + (2.39 + 0.509i)T + (10.0 + 4.47i)T^{2}
13 1+(−1.74+0.371i)T+(11.8−5.28i)T2 1 + (-1.74 + 0.371i)T + (11.8 - 5.28i)T^{2}
17 1+(−3.67−2.66i)T+(5.25+16.1i)T2 1 + (-3.67 - 2.66i)T + (5.25 + 16.1i)T^{2}
19 1+(−5.85−4.25i)T+(5.87+18.0i)T2 1 + (-5.85 - 4.25i)T + (5.87 + 18.0i)T^{2}
23 1+(−1.58+1.75i)T+(−2.40−22.8i)T2 1 + (-1.58 + 1.75i)T + (-2.40 - 22.8i)T^{2}
29 1+(−0.855−8.13i)T+(−28.3+6.02i)T2 1 + (-0.855 - 8.13i)T + (-28.3 + 6.02i)T^{2}
31 1+(0.709−6.74i)T+(−30.3−6.44i)T2 1 + (0.709 - 6.74i)T + (-30.3 - 6.44i)T^{2}
37 1+(−1.53+4.73i)T+(−29.9−21.7i)T2 1 + (-1.53 + 4.73i)T + (-29.9 - 21.7i)T^{2}
41 1+(−4.68+0.995i)T+(37.4−16.6i)T2 1 + (-4.68 + 0.995i)T + (37.4 - 16.6i)T^{2}
43 1+(−4.14+7.17i)T+(−21.5−37.2i)T2 1 + (-4.14 + 7.17i)T + (-21.5 - 37.2i)T^{2}
47 1+(−1.05−10.0i)T+(−45.9+9.77i)T2 1 + (-1.05 - 10.0i)T + (-45.9 + 9.77i)T^{2}
53 1+(2.52−1.83i)T+(16.3−50.4i)T2 1 + (2.52 - 1.83i)T + (16.3 - 50.4i)T^{2}
59 1+(10.2−2.17i)T+(53.8−23.9i)T2 1 + (10.2 - 2.17i)T + (53.8 - 23.9i)T^{2}
61 1+(−2.57−0.548i)T+(55.7+24.8i)T2 1 + (-2.57 - 0.548i)T + (55.7 + 24.8i)T^{2}
67 1+(−0.170+1.62i)T+(−65.5−13.9i)T2 1 + (-0.170 + 1.62i)T + (-65.5 - 13.9i)T^{2}
71 1+(−8.24+5.98i)T+(21.9−67.5i)T2 1 + (-8.24 + 5.98i)T + (21.9 - 67.5i)T^{2}
73 1+(0.608+1.87i)T+(−59.0+42.9i)T2 1 + (0.608 + 1.87i)T + (-59.0 + 42.9i)T^{2}
79 1+(1.55+14.8i)T+(−77.2+16.4i)T2 1 + (1.55 + 14.8i)T + (-77.2 + 16.4i)T^{2}
83 1+(−3.70+1.65i)T+(55.5−61.6i)T2 1 + (-3.70 + 1.65i)T + (55.5 - 61.6i)T^{2}
89 1+(−0.105−0.326i)T+(−72.0+52.3i)T2 1 + (-0.105 - 0.326i)T + (-72.0 + 52.3i)T^{2}
97 1+(−1.15−10.9i)T+(−94.8+20.1i)T2 1 + (-1.15 - 10.9i)T + (-94.8 + 20.1i)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

−12.25111556557661049973649842053, −10.79499701896827361778627809002, −10.37264379734685469245952919670, −9.362989314057613446765116276665, −8.885997463423283558030770654234, −7.67510081974011972612317841798, −6.17140885690663740883443589922, −5.34249557875273612518377990678, −3.13801780513246908978135179245, −1.24924017804354877153219695777, 0.905369034657321267552485791033, 2.57290968824194517097890269377, 5.48882783992019169058045351198, 6.46974501799522182227595865812, 7.30442377893333359271848317523, 7.894604997789627436702431697799, 9.572998604859615837044511625731, 9.979180100663370330183351105602, 11.02571109590368636374857653269, 11.57480147460216937370261901147

Graph of the ZZ-function along the critical line