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Properties

Label 2-507-507.227-c1-0-2
Degree 22
Conductor 507507
Sign −0.652+0.757i-0.652 + 0.757i
Analytic cond. 4.048414.04841
Root an. cond. 2.012062.01206
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  + (−0.481 + 1.66i)3-s + (−0.160 + 1.99i)4-s + (−3.64 − 1.46i)7-s + (−2.53 − 1.60i)9-s + (−3.23 − 1.22i)12-s + (−2.59 − 2.5i)13-s + (−3.94 − 0.641i)16-s + (5.49 + 1.47i)19-s + (4.19 − 5.35i)21-s + (−3.31 + 3.74i)25-s + (3.88 − 3.44i)27-s + (3.51 − 7.03i)28-s + (0.438 − 7.24i)31-s + (3.60 − 4.79i)36-s + (−6.69 + 10.1i)37-s + ⋯
L(s)  = 1  + (−0.278 + 0.960i)3-s + (−0.0804 + 0.996i)4-s + (−1.37 − 0.554i)7-s + (−0.845 − 0.534i)9-s + (−0.935 − 0.354i)12-s + (−0.720 − 0.693i)13-s + (−0.987 − 0.160i)16-s + (1.26 + 0.337i)19-s + (0.915 − 1.16i)21-s + (−0.663 + 0.748i)25-s + (0.748 − 0.663i)27-s + (0.663 − 1.32i)28-s + (0.0786 − 1.30i)31-s + (0.600 − 0.799i)36-s + (−1.10 + 1.66i)37-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.652+0.757i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.652+0.757i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.652+0.757i-0.652 + 0.757i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(227,⋅)\chi_{507} (227, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), −0.652+0.757i)(2,\ 507,\ (\ :1/2),\ -0.652 + 0.757i)

Particular Values

L(1)L(1) ≈\approx 0.0698747−0.152459i0.0698747 - 0.152459i
L(12)L(\frac12) ≈\approx 0.0698747−0.152459i0.0698747 - 0.152459i
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+(0.481−1.66i)T 1 + (0.481 - 1.66i)T
13 1+(2.59+2.5i)T 1 + (2.59 + 2.5i)T
good2 1+(0.160−1.99i)T2 1 + (0.160 - 1.99i)T^{2}
5 1+(3.31−3.74i)T2 1 + (3.31 - 3.74i)T^{2}
7 1+(3.64+1.46i)T+(5.04+4.84i)T2 1 + (3.64 + 1.46i)T + (5.04 + 4.84i)T^{2}
11 1+(−9.93−4.71i)T2 1 + (-9.93 - 4.71i)T^{2}
17 1+(11.7−12.2i)T2 1 + (11.7 - 12.2i)T^{2}
19 1+(−5.49−1.47i)T+(16.4+9.5i)T2 1 + (-5.49 - 1.47i)T + (16.4 + 9.5i)T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1+(28.9+2.33i)T2 1 + (28.9 + 2.33i)T^{2}
31 1+(−0.438+7.24i)T+(−30.7−3.73i)T2 1 + (-0.438 + 7.24i)T + (-30.7 - 3.73i)T^{2}
37 1+(6.69−10.1i)T+(−14.5−34.0i)T2 1 + (6.69 - 10.1i)T + (-14.5 - 34.0i)T^{2}
41 1+(21.9−34.6i)T2 1 + (21.9 - 34.6i)T^{2}
43 1+(12.4+2.53i)T+(39.5+16.8i)T2 1 + (12.4 + 2.53i)T + (39.5 + 16.8i)T^{2}
47 1+(43.9+16.6i)T2 1 + (43.9 + 16.6i)T^{2}
53 1+(51.4+12.6i)T2 1 + (51.4 + 12.6i)T^{2}
59 1+(−18.6−55.9i)T2 1 + (-18.6 - 55.9i)T^{2}
61 1+(12.4−9.38i)T+(16.9−58.5i)T2 1 + (12.4 - 9.38i)T + (16.9 - 58.5i)T^{2}
67 1+(6.63−7.80i)T+(−10.7−66.1i)T2 1 + (6.63 - 7.80i)T + (-10.7 - 66.1i)T^{2}
71 1+(−70.9−2.85i)T2 1 + (-70.9 - 2.85i)T^{2}
73 1+(1.60+5.15i)T+(−60.0+41.4i)T2 1 + (1.60 + 5.15i)T + (-60.0 + 41.4i)T^{2}
79 1+(8.45+12.2i)T+(−28.0+73.8i)T2 1 + (8.45 + 12.2i)T + (-28.0 + 73.8i)T^{2}
83 1+(−38.5+73.4i)T2 1 + (-38.5 + 73.4i)T^{2}
89 1+(−77.0+44.5i)T2 1 + (-77.0 + 44.5i)T^{2}
97 1+(−1.70−16.9i)T+(−95.0+19.4i)T2 1 + (-1.70 - 16.9i)T + (-95.0 + 19.4i)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

−11.65163634753640032956255229513, −10.29680262719255646570341832920, −9.835000635003275579012786671995, −9.016474076465623049154031638502, −7.85668273828177456663919338978, −6.99589469644383244489053920336, −5.85977898410385106289546796707, −4.71098762333120817415232483426, −3.52632418158682961006849401652, −3.06778052177355812319489869608, 0.097337389466126812515423586290, 1.83616184795605944486403103495, 3.08767748715571094108039965021, 4.94322852684127128953648988569, 5.82474195603312589455397281499, 6.61245367402334725176013945411, 7.27568816860675624004811438676, 8.726620807134414795493618051986, 9.509714092026067525382359702299, 10.23432719980748448688014846121

Graph of the ZZ-function along the critical line