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Properties

Label 2-507-39.20-c1-0-23
Degree 22
Conductor 507507
Sign 0.533−0.846i0.533 - 0.846i
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  + (0.619 + 2.31i)2-s + (0.866 − 1.5i)3-s + (−3.23 + 1.86i)4-s + (1.23 − 1.23i)5-s + (4.00 + 1.07i)6-s + (−2.93 − 2.93i)8-s + (−1.5 − 2.59i)9-s + (3.63 + 2.09i)10-s + (6.31 − 1.69i)11-s + 6.46i·12-s + (−0.785 − 2.93i)15-s + (1.23 − 2.13i)16-s + (5.07 − 5.07i)18-s + (−1.69 + 6.31i)20-s + (7.83 + 13.5i)22-s + ⋯
L(s)  = 1  + (0.438 + 1.63i)2-s + (0.499 − 0.866i)3-s + (−1.61 + 0.933i)4-s + (0.554 − 0.554i)5-s + (1.63 + 0.438i)6-s + (−1.03 − 1.03i)8-s + (−0.5 − 0.866i)9-s + (1.14 + 0.663i)10-s + (1.90 − 0.510i)11-s + 1.86i·12-s + (−0.202 − 0.757i)15-s + (0.308 − 0.533i)16-s + (1.19 − 1.19i)18-s + (−0.378 + 1.41i)20-s + (1.66 + 2.89i)22-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((0.533−0.846i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((0.533−0.846i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 0.533−0.846i0.533 - 0.846i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(488,⋅)\chi_{507} (488, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), 0.533−0.846i)(2,\ 507,\ (\ :1/2),\ 0.533 - 0.846i)

Particular Values

L(1)L(1) ≈\approx 1.94765+1.07488i1.94765 + 1.07488i
L(12)L(\frac12) ≈\approx 1.94765+1.07488i1.94765 + 1.07488i
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
13 1 1
good2 1+(−0.619−2.31i)T+(−1.73+i)T2 1 + (-0.619 - 2.31i)T + (-1.73 + i)T^{2}
5 1+(−1.23+1.23i)T−5iT2 1 + (-1.23 + 1.23i)T - 5iT^{2}
7 1+(6.06+3.5i)T2 1 + (6.06 + 3.5i)T^{2}
11 1+(−6.31+1.69i)T+(9.52−5.5i)T2 1 + (-6.31 + 1.69i)T + (9.52 - 5.5i)T^{2}
17 1+(−8.5+14.7i)T2 1 + (-8.5 + 14.7i)T^{2}
19 1+(−16.4−9.5i)T2 1 + (-16.4 - 9.5i)T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1+(14.5+25.1i)T2 1 + (14.5 + 25.1i)T^{2}
31 1+31iT2 1 + 31iT^{2}
37 1+(−32.0+18.5i)T2 1 + (-32.0 + 18.5i)T^{2}
41 1+(−2.02−7.55i)T+(−35.5+20.5i)T2 1 + (-2.02 - 7.55i)T + (-35.5 + 20.5i)T^{2}
43 1+(3.46−2i)T+(21.5−37.2i)T2 1 + (3.46 - 2i)T + (21.5 - 37.2i)T^{2}
47 1+(−7.10−7.10i)T+47iT2 1 + (-7.10 - 7.10i)T + 47iT^{2}
53 1−53T2 1 - 53T^{2}
59 1+(−0.121+0.453i)T+(−51.0−29.5i)T2 1 + (-0.121 + 0.453i)T + (-51.0 - 29.5i)T^{2}
61 1+(6.92+12i)T+(−30.5+52.8i)T2 1 + (6.92 + 12i)T + (-30.5 + 52.8i)T^{2}
67 1+(58.0−33.5i)T2 1 + (58.0 - 33.5i)T^{2}
71 1+(15.5+4.17i)T+(61.4+35.5i)T2 1 + (15.5 + 4.17i)T + (61.4 + 35.5i)T^{2}
73 1−73iT2 1 - 73iT^{2}
79 1+10.3T+79T2 1 + 10.3T + 79T^{2}
83 1+(−8.91+8.91i)T−83iT2 1 + (-8.91 + 8.91i)T - 83iT^{2}
89 1+(4.17−1.11i)T+(77.0−44.5i)T2 1 + (4.17 - 1.11i)T + (77.0 - 44.5i)T^{2}
97 1+(−84.0−48.5i)T2 1 + (-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

−11.35287882473406867547404732002, −9.450604904442609960410433291819, −8.996296957172584094836810357017, −8.194930060343153048645136338328, −7.26396410043501183134563452619, −6.37152749939267286425864190600, −5.93736053160599651970331953924, −4.63551374383456508573874680378, −3.46883369951088156917398547757, −1.43735324962412166409103779611, 1.69938318644832911656625623526, 2.76756259704859255798247468964, 3.83117910821008392603431391658, 4.45430129627352214445097803791, 5.75121170600826549695010430161, 7.04774918222428956665985648111, 8.747841097633110119903713886069, 9.357224695980054375459852789892, 10.09185580617744469236840519063, 10.66675318433464831092975500659

Graph of the ZZ-function along the critical line