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Properties

Label 2-15e2-45.4-c1-0-7
Degree 22
Conductor 225225
Sign 0.917+0.397i0.917 + 0.397i
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  + (−0.866 + 0.5i)2-s + (0.866 − 1.5i)3-s + (−0.500 + 0.866i)4-s + 1.73i·6-s + (2.59 − 1.5i)7-s − 3i·8-s + (−1.5 − 2.59i)9-s + (1 + 1.73i)11-s + (0.866 + 1.5i)12-s + (1.73 + i)13-s + (−1.5 + 2.59i)14-s + (0.500 + 0.866i)16-s − 4i·17-s + (2.59 + 1.5i)18-s + 8·19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.499 − 0.866i)3-s + (−0.250 + 0.433i)4-s + 0.707i·6-s + (0.981 − 0.566i)7-s − 1.06i·8-s + (−0.5 − 0.866i)9-s + (0.301 + 0.522i)11-s + (0.249 + 0.433i)12-s + (0.480 + 0.277i)13-s + (−0.400 + 0.694i)14-s + (0.125 + 0.216i)16-s − 0.970i·17-s + (0.612 + 0.353i)18-s + 1.83·19-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((0.917+0.397i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((0.917+0.397i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: 0.917+0.397i0.917 + 0.397i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ225(49,⋅)\chi_{225} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 225, ( :1/2), 0.917+0.397i)(2,\ 225,\ (\ :1/2),\ 0.917 + 0.397i)

Particular Values

L(1)L(1) ≈\approx 1.04822−0.217409i1.04822 - 0.217409i
L(12)L(\frac12) ≈\approx 1.04822−0.217409i1.04822 - 0.217409i
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+(−0.866+1.5i)T 1 + (-0.866 + 1.5i)T
5 1 1
good2 1+(0.866−0.5i)T+(1−1.73i)T2 1 + (0.866 - 0.5i)T + (1 - 1.73i)T^{2}
7 1+(−2.59+1.5i)T+(3.5−6.06i)T2 1 + (-2.59 + 1.5i)T + (3.5 - 6.06i)T^{2}
11 1+(−1−1.73i)T+(−5.5+9.52i)T2 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2}
13 1+(−1.73−i)T+(6.5+11.2i)T2 1 + (-1.73 - i)T + (6.5 + 11.2i)T^{2}
17 1+4iT−17T2 1 + 4iT - 17T^{2}
19 1−8T+19T2 1 - 8T + 19T^{2}
23 1+(2.59+1.5i)T+(11.5+19.9i)T2 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2}
29 1+(0.5+0.866i)T+(−14.5+25.1i)T2 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2}
31 1+(−15.5−26.8i)T2 1 + (-15.5 - 26.8i)T^{2}
37 1−4iT−37T2 1 - 4iT - 37T^{2}
41 1+(2.5−4.33i)T+(−20.5−35.5i)T2 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2}
43 1+(6.92−4i)T+(21.5−37.2i)T2 1 + (6.92 - 4i)T + (21.5 - 37.2i)T^{2}
47 1+(6.06−3.5i)T+(23.5−40.7i)T2 1 + (6.06 - 3.5i)T + (23.5 - 40.7i)T^{2}
53 1+2iT−53T2 1 + 2iT - 53T^{2}
59 1+(7−12.1i)T+(−29.5−51.0i)T2 1 + (7 - 12.1i)T + (-29.5 - 51.0i)T^{2}
61 1+(3.5+6.06i)T+(−30.5+52.8i)T2 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.59+1.5i)T+(33.5+58.0i)T2 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2}
71 1−2T+71T2 1 - 2T + 71T^{2}
73 1−4iT−73T2 1 - 4iT - 73T^{2}
79 1+(3+5.19i)T+(−39.5+68.4i)T2 1 + (3 + 5.19i)T + (-39.5 + 68.4i)T^{2}
83 1+(−7.79+4.5i)T+(41.5−71.8i)T2 1 + (-7.79 + 4.5i)T + (41.5 - 71.8i)T^{2}
89 1−15T+89T2 1 - 15T + 89T^{2}
97 1+(1.73−i)T+(48.5−84.0i)T2 1 + (1.73 - i)T + (48.5 - 84.0i)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

−12.07802479609158136582253854710, −11.50227629745020673884248933243, −9.864549335628471389833744011795, −9.062248999002875606993402376261, −7.968780208217688930806018493554, −7.51386289938755394923643911101, −6.53159072992239292963102843242, −4.70770831648353199562520420336, −3.27877565443137169676674328242, −1.29092685980446188214799950289, 1.77190946552616613835539648406, 3.49208418543190910432014017075, 5.01176078281149394015673956521, 5.75790584858429441523721573098, 7.912352711906511156333769662445, 8.578638647989805930346787557363, 9.375817695707002706557743251260, 10.30334993026600208030861775769, 11.12804855227929657346593998576, 11.84122935949544078252744170935

Graph of the ZZ-function along the critical line