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Properties

Label 2-15e2-225.106-c1-0-3
Degree 22
Conductor 225225
Sign −0.758−0.651i-0.758 - 0.651i
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  + (0.429 + 0.0913i)2-s + (0.335 + 1.69i)3-s + (−1.65 − 0.735i)4-s + (−1.76 + 1.36i)5-s + (−0.0109 + 0.760i)6-s + (−0.888 + 1.53i)7-s + (−1.35 − 0.982i)8-s + (−2.77 + 1.14i)9-s + (−0.885 + 0.425i)10-s + (4.21 + 0.895i)11-s + (0.694 − 3.05i)12-s + (−6.27 + 1.33i)13-s + (−0.522 + 0.579i)14-s + (−2.91 − 2.54i)15-s + (1.92 + 2.14i)16-s + (0.162 + 0.118i)17-s + ⋯
L(s)  = 1  + (0.303 + 0.0645i)2-s + (0.193 + 0.981i)3-s + (−0.825 − 0.367i)4-s + (−0.791 + 0.611i)5-s + (−0.00446 + 0.310i)6-s + (−0.335 + 0.581i)7-s + (−0.478 − 0.347i)8-s + (−0.924 + 0.380i)9-s + (−0.279 + 0.134i)10-s + (1.26 + 0.269i)11-s + (0.200 − 0.881i)12-s + (−1.74 + 0.369i)13-s + (−0.139 + 0.154i)14-s + (−0.753 − 0.657i)15-s + (0.481 + 0.535i)16-s + (0.0395 + 0.0287i)17-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((−0.758−0.651i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((−0.758−0.651i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: −0.758−0.651i-0.758 - 0.651i
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), −0.758−0.651i)(2,\ 225,\ (\ :1/2),\ -0.758 - 0.651i)

Particular Values

L(1)L(1) ≈\approx 0.286334+0.772267i0.286334 + 0.772267i
L(12)L(\frac12) ≈\approx 0.286334+0.772267i0.286334 + 0.772267i
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+(−0.335−1.69i)T 1 + (-0.335 - 1.69i)T
5 1+(1.76−1.36i)T 1 + (1.76 - 1.36i)T
good2 1+(−0.429−0.0913i)T+(1.82+0.813i)T2 1 + (-0.429 - 0.0913i)T + (1.82 + 0.813i)T^{2}
7 1+(0.888−1.53i)T+(−3.5−6.06i)T2 1 + (0.888 - 1.53i)T + (-3.5 - 6.06i)T^{2}
11 1+(−4.21−0.895i)T+(10.0+4.47i)T2 1 + (-4.21 - 0.895i)T + (10.0 + 4.47i)T^{2}
13 1+(6.27−1.33i)T+(11.8−5.28i)T2 1 + (6.27 - 1.33i)T + (11.8 - 5.28i)T^{2}
17 1+(−0.162−0.118i)T+(5.25+16.1i)T2 1 + (-0.162 - 0.118i)T + (5.25 + 16.1i)T^{2}
19 1+(−4.79−3.48i)T+(5.87+18.0i)T2 1 + (-4.79 - 3.48i)T + (5.87 + 18.0i)T^{2}
23 1+(−1.83+2.03i)T+(−2.40−22.8i)T2 1 + (-1.83 + 2.03i)T + (-2.40 - 22.8i)T^{2}
29 1+(−0.0773−0.735i)T+(−28.3+6.02i)T2 1 + (-0.0773 - 0.735i)T + (-28.3 + 6.02i)T^{2}
31 1+(0.289−2.75i)T+(−30.3−6.44i)T2 1 + (0.289 - 2.75i)T + (-30.3 - 6.44i)T^{2}
37 1+(1.36−4.21i)T+(−29.9−21.7i)T2 1 + (1.36 - 4.21i)T + (-29.9 - 21.7i)T^{2}
41 1+(−12.3+2.61i)T+(37.4−16.6i)T2 1 + (-12.3 + 2.61i)T + (37.4 - 16.6i)T^{2}
43 1+(3.91−6.78i)T+(−21.5−37.2i)T2 1 + (3.91 - 6.78i)T + (-21.5 - 37.2i)T^{2}
47 1+(0.0518+0.493i)T+(−45.9+9.77i)T2 1 + (0.0518 + 0.493i)T + (-45.9 + 9.77i)T^{2}
53 1+(−5.99+4.35i)T+(16.3−50.4i)T2 1 + (-5.99 + 4.35i)T + (16.3 - 50.4i)T^{2}
59 1+(9.52−2.02i)T+(53.8−23.9i)T2 1 + (9.52 - 2.02i)T + (53.8 - 23.9i)T^{2}
61 1+(9.51+2.02i)T+(55.7+24.8i)T2 1 + (9.51 + 2.02i)T + (55.7 + 24.8i)T^{2}
67 1+(0.0149−0.141i)T+(−65.5−13.9i)T2 1 + (0.0149 - 0.141i)T + (-65.5 - 13.9i)T^{2}
71 1+(−1.95+1.41i)T+(21.9−67.5i)T2 1 + (-1.95 + 1.41i)T + (21.9 - 67.5i)T^{2}
73 1+(0.860+2.64i)T+(−59.0+42.9i)T2 1 + (0.860 + 2.64i)T + (-59.0 + 42.9i)T^{2}
79 1+(1.34+12.7i)T+(−77.2+16.4i)T2 1 + (1.34 + 12.7i)T + (-77.2 + 16.4i)T^{2}
83 1+(−3.13+1.39i)T+(55.5−61.6i)T2 1 + (-3.13 + 1.39i)T + (55.5 - 61.6i)T^{2}
89 1+(−3.29−10.1i)T+(−72.0+52.3i)T2 1 + (-3.29 - 10.1i)T + (-72.0 + 52.3i)T^{2}
97 1+(−1.65−15.7i)T+(−94.8+20.1i)T2 1 + (-1.65 - 15.7i)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

−12.31404372636336647288160682058, −11.86132489397014026806398753040, −10.51131399911384077869698335501, −9.562845726982192665503596507479, −9.088588987982817377872862066349, −7.71383149734483438350758336359, −6.34101877452598431385546924339, −5.02634211347520659503166412017, −4.14388478769693578212847889894, −3.03523582772307085497534197825, 0.63382424947647006242713611932, 3.09266965104391091818712969343, 4.25972730310862628328875707065, 5.47710736342166051376945518458, 7.15548360030558150758718569455, 7.68556126318465741316015865898, 8.931186802447378671827980162404, 9.532812416276354449331812656994, 11.42414143082954279569235176885, 12.14340421107266642983958845414

Graph of the ZZ-function along the critical line