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Properties

Label 2-459-153.106-c1-0-11
Degree 22
Conductor 459459
Sign 0.996+0.0856i0.996 + 0.0856i
Analytic cond. 3.665133.66513
Root an. cond. 1.914451.91445
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  + (0.793 + 0.458i)2-s + (−0.580 − 1.00i)4-s + (3.41 + 0.915i)5-s + (−0.340 + 0.0913i)7-s − 2.89i·8-s + (2.29 + 2.29i)10-s + (3.14 − 0.843i)11-s + (−2.19 − 3.79i)13-s + (−0.312 − 0.0836i)14-s + (0.165 − 0.286i)16-s + (−0.593 + 4.08i)17-s + 3.53i·19-s + (−1.06 − 3.96i)20-s + (2.88 + 0.772i)22-s + (1.54 − 5.78i)23-s + ⋯
L(s)  = 1  + (0.560 + 0.323i)2-s + (−0.290 − 0.502i)4-s + (1.52 + 0.409i)5-s + (−0.128 + 0.0345i)7-s − 1.02i·8-s + (0.724 + 0.724i)10-s + (0.948 − 0.254i)11-s + (−0.607 − 1.05i)13-s + (−0.0834 − 0.0223i)14-s + (0.0412 − 0.0715i)16-s + (−0.144 + 0.989i)17-s + 0.810i·19-s + (−0.237 − 0.886i)20-s + (0.614 + 0.164i)22-s + (0.323 − 1.20i)23-s + ⋯

Functional equation

Λ(s)=(459s/2 ΓC(s) L(s)=((0.996+0.0856i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0856i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(459s/2 ΓC(s+1/2) L(s)=((0.996+0.0856i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0856i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 459459    =    33⋅173^{3} \cdot 17
Sign: 0.996+0.0856i0.996 + 0.0856i
Analytic conductor: 3.665133.66513
Root analytic conductor: 1.914451.91445
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ459(208,⋅)\chi_{459} (208, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 459, ( :1/2), 0.996+0.0856i)(2,\ 459,\ (\ :1/2),\ 0.996 + 0.0856i)

Particular Values

L(1)L(1) ≈\approx 2.17575−0.0933474i2.17575 - 0.0933474i
L(12)L(\frac12) ≈\approx 2.17575−0.0933474i2.17575 - 0.0933474i
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
17 1+(0.593−4.08i)T 1 + (0.593 - 4.08i)T
good2 1+(−0.793−0.458i)T+(1+1.73i)T2 1 + (-0.793 - 0.458i)T + (1 + 1.73i)T^{2}
5 1+(−3.41−0.915i)T+(4.33+2.5i)T2 1 + (-3.41 - 0.915i)T + (4.33 + 2.5i)T^{2}
7 1+(0.340−0.0913i)T+(6.06−3.5i)T2 1 + (0.340 - 0.0913i)T + (6.06 - 3.5i)T^{2}
11 1+(−3.14+0.843i)T+(9.52−5.5i)T2 1 + (-3.14 + 0.843i)T + (9.52 - 5.5i)T^{2}
13 1+(2.19+3.79i)T+(−6.5+11.2i)T2 1 + (2.19 + 3.79i)T + (-6.5 + 11.2i)T^{2}
19 1−3.53iT−19T2 1 - 3.53iT - 19T^{2}
23 1+(−1.54+5.78i)T+(−19.9−11.5i)T2 1 + (-1.54 + 5.78i)T + (-19.9 - 11.5i)T^{2}
29 1+(−0.880−3.28i)T+(−25.1+14.5i)T2 1 + (-0.880 - 3.28i)T + (-25.1 + 14.5i)T^{2}
31 1+(−6.39−1.71i)T+(26.8+15.5i)T2 1 + (-6.39 - 1.71i)T + (26.8 + 15.5i)T^{2}
37 1+(1.16−1.16i)T−37iT2 1 + (1.16 - 1.16i)T - 37iT^{2}
41 1+(−1.59+5.95i)T+(−35.5−20.5i)T2 1 + (-1.59 + 5.95i)T + (-35.5 - 20.5i)T^{2}
43 1+(4.98+2.87i)T+(21.5+37.2i)T2 1 + (4.98 + 2.87i)T + (21.5 + 37.2i)T^{2}
47 1+(1.45−2.51i)T+(−23.5−40.7i)T2 1 + (1.45 - 2.51i)T + (-23.5 - 40.7i)T^{2}
53 1−11.0iT−53T2 1 - 11.0iT - 53T^{2}
59 1+(−2.82+1.62i)T+(29.5−51.0i)T2 1 + (-2.82 + 1.62i)T + (29.5 - 51.0i)T^{2}
61 1+(4.30−1.15i)T+(52.8−30.5i)T2 1 + (4.30 - 1.15i)T + (52.8 - 30.5i)T^{2}
67 1+(5.68+9.84i)T+(−33.5+58.0i)T2 1 + (5.68 + 9.84i)T + (-33.5 + 58.0i)T^{2}
71 1+(3.98−3.98i)T−71iT2 1 + (3.98 - 3.98i)T - 71iT^{2}
73 1+(10.0−10.0i)T−73iT2 1 + (10.0 - 10.0i)T - 73iT^{2}
79 1+(10.7−2.88i)T+(68.4−39.5i)T2 1 + (10.7 - 2.88i)T + (68.4 - 39.5i)T^{2}
83 1+(3.56+2.05i)T+(41.5+71.8i)T2 1 + (3.56 + 2.05i)T + (41.5 + 71.8i)T^{2}
89 1−1.04T+89T2 1 - 1.04T + 89T^{2}
97 1+(−1.26−4.73i)T+(−84.0+48.5i)T2 1 + (-1.26 - 4.73i)T + (-84.0 + 48.5i)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

−10.58451675297791856152567279518, −10.27474465999293700907945852064, −9.422383637398046851902062056465, −8.485008909086342260566254969802, −6.89277274403206866523261800611, −6.15935656768773145429534765344, −5.59997283986505260155478654743, −4.44930020511182834720561086163, −3.01707269408466727180322586996, −1.45585544602072140026150270703, 1.79363443633336014439636301581, 2.92413384414537863941989564852, 4.43682922561133898465165192739, 5.09360205898992430058452561122, 6.29056125133981738893007985250, 7.22653159007599022551830687936, 8.665747831704375597574333569076, 9.428307670877771941860172745151, 9.857915720300607951541773266275, 11.56501271069805686489849497478

Graph of the ZZ-function along the critical line