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Properties

Label 2-15e2-225.106-c1-0-8
Degree 22
Conductor 225225
Sign −0.0718+0.997i-0.0718 + 0.997i
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  + (−2.27 − 0.484i)2-s + (−1.66 − 0.491i)3-s + (3.13 + 1.39i)4-s + (−1.74 + 1.39i)5-s + (3.54 + 1.92i)6-s + (−0.0399 + 0.0692i)7-s + (−2.69 − 1.95i)8-s + (2.51 + 1.63i)9-s + (4.65 − 2.33i)10-s + (0.117 + 0.0249i)11-s + (−4.51 − 3.85i)12-s + (0.896 − 0.190i)13-s + (0.124 − 0.138i)14-s + (3.58 − 1.45i)15-s + (0.601 + 0.668i)16-s + (−2.65 − 1.92i)17-s + ⋯
L(s)  = 1  + (−1.61 − 0.342i)2-s + (−0.958 − 0.283i)3-s + (1.56 + 0.697i)4-s + (−0.781 + 0.624i)5-s + (1.44 + 0.785i)6-s + (−0.0151 + 0.0261i)7-s + (−0.952 − 0.691i)8-s + (0.838 + 0.544i)9-s + (1.47 − 0.738i)10-s + (0.0353 + 0.00751i)11-s + (−1.30 − 1.11i)12-s + (0.248 − 0.0528i)13-s + (0.0333 − 0.0370i)14-s + (0.926 − 0.376i)15-s + (0.150 + 0.167i)16-s + (−0.643 − 0.467i)17-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((−0.0718+0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0718 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((−0.0718+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0718 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: −0.0718+0.997i-0.0718 + 0.997i
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), −0.0718+0.997i)(2,\ 225,\ (\ :1/2),\ -0.0718 + 0.997i)

Particular Values

L(1)L(1) ≈\approx 0.182426−0.196043i0.182426 - 0.196043i
L(12)L(\frac12) ≈\approx 0.182426−0.196043i0.182426 - 0.196043i
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+(1.66+0.491i)T 1 + (1.66 + 0.491i)T
5 1+(1.74−1.39i)T 1 + (1.74 - 1.39i)T
good2 1+(2.27+0.484i)T+(1.82+0.813i)T2 1 + (2.27 + 0.484i)T + (1.82 + 0.813i)T^{2}
7 1+(0.0399−0.0692i)T+(−3.5−6.06i)T2 1 + (0.0399 - 0.0692i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.117−0.0249i)T+(10.0+4.47i)T2 1 + (-0.117 - 0.0249i)T + (10.0 + 4.47i)T^{2}
13 1+(−0.896+0.190i)T+(11.8−5.28i)T2 1 + (-0.896 + 0.190i)T + (11.8 - 5.28i)T^{2}
17 1+(2.65+1.92i)T+(5.25+16.1i)T2 1 + (2.65 + 1.92i)T + (5.25 + 16.1i)T^{2}
19 1+(2.09+1.51i)T+(5.87+18.0i)T2 1 + (2.09 + 1.51i)T + (5.87 + 18.0i)T^{2}
23 1+(−4.80+5.33i)T+(−2.40−22.8i)T2 1 + (-4.80 + 5.33i)T + (-2.40 - 22.8i)T^{2}
29 1+(0.650+6.18i)T+(−28.3+6.02i)T2 1 + (0.650 + 6.18i)T + (-28.3 + 6.02i)T^{2}
31 1+(−0.644+6.13i)T+(−30.3−6.44i)T2 1 + (-0.644 + 6.13i)T + (-30.3 - 6.44i)T^{2}
37 1+(3.64−11.2i)T+(−29.9−21.7i)T2 1 + (3.64 - 11.2i)T + (-29.9 - 21.7i)T^{2}
41 1+(−6.21+1.32i)T+(37.4−16.6i)T2 1 + (-6.21 + 1.32i)T + (37.4 - 16.6i)T^{2}
43 1+(−6.13+10.6i)T+(−21.5−37.2i)T2 1 + (-6.13 + 10.6i)T + (-21.5 - 37.2i)T^{2}
47 1+(−0.401−3.81i)T+(−45.9+9.77i)T2 1 + (-0.401 - 3.81i)T + (-45.9 + 9.77i)T^{2}
53 1+(−6.52+4.73i)T+(16.3−50.4i)T2 1 + (-6.52 + 4.73i)T + (16.3 - 50.4i)T^{2}
59 1+(1.33−0.284i)T+(53.8−23.9i)T2 1 + (1.33 - 0.284i)T + (53.8 - 23.9i)T^{2}
61 1+(9.81+2.08i)T+(55.7+24.8i)T2 1 + (9.81 + 2.08i)T + (55.7 + 24.8i)T^{2}
67 1+(−0.885+8.42i)T+(−65.5−13.9i)T2 1 + (-0.885 + 8.42i)T + (-65.5 - 13.9i)T^{2}
71 1+(−5.54+4.02i)T+(21.9−67.5i)T2 1 + (-5.54 + 4.02i)T + (21.9 - 67.5i)T^{2}
73 1+(−1.36−4.20i)T+(−59.0+42.9i)T2 1 + (-1.36 - 4.20i)T + (-59.0 + 42.9i)T^{2}
79 1+(0.223+2.12i)T+(−77.2+16.4i)T2 1 + (0.223 + 2.12i)T + (-77.2 + 16.4i)T^{2}
83 1+(−2.75+1.22i)T+(55.5−61.6i)T2 1 + (-2.75 + 1.22i)T + (55.5 - 61.6i)T^{2}
89 1+(4.33+13.3i)T+(−72.0+52.3i)T2 1 + (4.33 + 13.3i)T + (-72.0 + 52.3i)T^{2}
97 1+(1.27+12.1i)T+(−94.8+20.1i)T2 1 + (1.27 + 12.1i)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

−11.54508952116823658170718024842, −10.93969799470002079717158806909, −10.30459275355908297124219039775, −9.097672296286504505739664402863, −8.019066687057263545510728266879, −7.15420647912561557503599871188, −6.35009846553633379097506446708, −4.47983331336878564466369264759, −2.47343052114957686509362929410, −0.48998198338046678892037770887, 1.21766169777927125396581284609, 4.04227161965002617208355037170, 5.46328705354978341723206500079, 6.76575527452397781885724668147, 7.57258222774442878992314940247, 8.769223892391147790476814723697, 9.348415101934838874036554042825, 10.71334145010124179753064570172, 11.02520749983099396319188227429, 12.17686165636652656293430888615

Graph of the ZZ-function along the critical line