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Properties

Label 2-507-13.9-c1-0-5
Degree 22
Conductor 507507
Sign −0.872−0.488i-0.872 - 0.488i
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  + (−0.780 + 1.35i)2-s + (−0.5 + 0.866i)3-s + (−0.219 − 0.379i)4-s + 3.56·5-s + (−0.780 − 1.35i)6-s + (0.280 + 0.486i)7-s − 2.43·8-s + (−0.499 − 0.866i)9-s + (−2.78 + 4.81i)10-s + (−1 + 1.73i)11-s + 0.438·12-s − 0.876·14-s + (−1.78 + 3.08i)15-s + (2.34 − 4.05i)16-s + (0.780 + 1.35i)17-s + 1.56·18-s + ⋯
L(s)  = 1  + (−0.552 + 0.956i)2-s + (−0.288 + 0.499i)3-s + (−0.109 − 0.189i)4-s + 1.59·5-s + (−0.318 − 0.552i)6-s + (0.106 + 0.183i)7-s − 0.862·8-s + (−0.166 − 0.288i)9-s + (−0.879 + 1.52i)10-s + (−0.301 + 0.522i)11-s + 0.126·12-s − 0.234·14-s + (−0.459 + 0.796i)15-s + (0.585 − 1.01i)16-s + (0.189 + 0.327i)17-s + 0.368·18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.872−0.488i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.872−0.488i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.872−0.488i-0.872 - 0.488i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(22,⋅)\chi_{507} (22, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), −0.872−0.488i)(2,\ 507,\ (\ :1/2),\ -0.872 - 0.488i)

Particular Values

L(1)L(1) ≈\approx 0.303864+1.16383i0.303864 + 1.16383i
L(12)L(\frac12) ≈\approx 0.303864+1.16383i0.303864 + 1.16383i
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
13 1 1
good2 1+(0.780−1.35i)T+(−1−1.73i)T2 1 + (0.780 - 1.35i)T + (-1 - 1.73i)T^{2}
5 1−3.56T+5T2 1 - 3.56T + 5T^{2}
7 1+(−0.280−0.486i)T+(−3.5+6.06i)T2 1 + (-0.280 - 0.486i)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}
17 1+(−0.780−1.35i)T+(−8.5+14.7i)T2 1 + (-0.780 - 1.35i)T + (-8.5 + 14.7i)T^{2}
19 1+(−3.56−6.16i)T+(−9.5+16.4i)T2 1 + (-3.56 - 6.16i)T + (-9.5 + 16.4i)T^{2}
23 1+(1−1.73i)T+(−11.5−19.9i)T2 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2}
29 1+(3.34−5.78i)T+(−14.5−25.1i)T2 1 + (3.34 - 5.78i)T + (-14.5 - 25.1i)T^{2}
31 1+2.56T+31T2 1 + 2.56T + 31T^{2}
37 1+(−3.78+6.54i)T+(−18.5−32.0i)T2 1 + (-3.78 + 6.54i)T + (-18.5 - 32.0i)T^{2}
41 1+(0.780−1.35i)T+(−20.5−35.5i)T2 1 + (0.780 - 1.35i)T + (-20.5 - 35.5i)T^{2}
43 1+(2.28+3.95i)T+(−21.5+37.2i)T2 1 + (2.28 + 3.95i)T + (-21.5 + 37.2i)T^{2}
47 1+8.24T+47T2 1 + 8.24T + 47T^{2}
53 1+0.684T+53T2 1 + 0.684T + 53T^{2}
59 1+(1.43+2.49i)T+(−29.5+51.0i)T2 1 + (1.43 + 2.49i)T + (-29.5 + 51.0i)T^{2}
61 1+(1.93+3.35i)T+(−30.5+52.8i)T2 1 + (1.93 + 3.35i)T + (-30.5 + 52.8i)T^{2}
67 1+(−2.28+3.95i)T+(−33.5−58.0i)T2 1 + (-2.28 + 3.95i)T + (-33.5 - 58.0i)T^{2}
71 1+(−7−12.1i)T+(−35.5+61.4i)T2 1 + (-7 - 12.1i)T + (-35.5 + 61.4i)T^{2}
73 1−10.1T+73T2 1 - 10.1T + 73T^{2}
79 1−5.43T+79T2 1 - 5.43T + 79T^{2}
83 1−0.876T+83T2 1 - 0.876T + 83T^{2}
89 1+(−2.43+4.22i)T+(−44.5−77.0i)T2 1 + (-2.43 + 4.22i)T + (-44.5 - 77.0i)T^{2}
97 1+(4.28+7.41i)T+(−48.5+84.0i)T2 1 + (4.28 + 7.41i)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

−11.02714778780201579775249673847, −9.910217737095316981207919228059, −9.630125900394697373951662510743, −8.665724483267901869657165252036, −7.64651741656644496436944365778, −6.63576926055383295530009825877, −5.70555074024199559945781498270, −5.28989216872990357745754169094, −3.45110746090661217016964648624, −1.90846095971533647900261052386, 0.908229563814700896464000294106, 2.11054503878639634575854936905, 2.98623095044222681985502288703, 5.04955947583767624716138085244, 5.93630573018603626252572068759, 6.68061807172844379866689607232, 8.024859747302351842951245503901, 9.246886646932221049148530385469, 9.649124351971963655876216382090, 10.59166399526128512603339041308

Graph of the ZZ-function along the critical line