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Properties

Label 2-1352-13.4-c1-0-23
Degree 22
Conductor 13521352
Sign 0.0771+0.997i0.0771 + 0.997i
Analytic cond. 10.795710.7957
Root an. cond. 3.285693.28569
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  + (−0.5 + 0.866i)3-s − i·5-s + (−4.33 + 2.5i)7-s + (1 + 1.73i)9-s + (−1.73 − i)11-s + (0.866 + 0.5i)15-s + (−1.5 − 2.59i)17-s + (−1.73 + i)19-s − 5i·21-s + (2 − 3.46i)23-s + 4·25-s − 5·27-s + (3 − 5.19i)29-s − 4i·31-s + (1.73 − 0.999i)33-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s − 0.447i·5-s + (−1.63 + 0.944i)7-s + (0.333 + 0.577i)9-s + (−0.522 − 0.301i)11-s + (0.223 + 0.129i)15-s + (−0.363 − 0.630i)17-s + (−0.397 + 0.229i)19-s − 1.09i·21-s + (0.417 − 0.722i)23-s + 0.800·25-s − 0.962·27-s + (0.557 − 0.964i)29-s − 0.718i·31-s + (0.301 − 0.174i)33-s + ⋯

Functional equation

Λ(s)=(1352s/2 ΓC(s) L(s)=((0.0771+0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0771 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1352s/2 ΓC(s+1/2) L(s)=((0.0771+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0771 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13521352    =    23⋅1322^{3} \cdot 13^{2}
Sign: 0.0771+0.997i0.0771 + 0.997i
Analytic conductor: 10.795710.7957
Root analytic conductor: 3.285693.28569
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1352(1161,⋅)\chi_{1352} (1161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1352, ( :1/2), 0.0771+0.997i)(2,\ 1352,\ (\ :1/2),\ 0.0771 + 0.997i)

Particular Values

L(1)L(1) ≈\approx 0.53242496980.5324249698
L(12)L(\frac12) ≈\approx 0.53242496980.5324249698
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)
bad2 1 1
13 1 1
good3 1+(0.5−0.866i)T+(−1.5−2.59i)T2 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2}
5 1+iT−5T2 1 + iT - 5T^{2}
7 1+(4.33−2.5i)T+(3.5−6.06i)T2 1 + (4.33 - 2.5i)T + (3.5 - 6.06i)T^{2}
11 1+(1.73+i)T+(5.5+9.52i)T2 1 + (1.73 + i)T + (5.5 + 9.52i)T^{2}
17 1+(1.5+2.59i)T+(−8.5+14.7i)T2 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.73−i)T+(9.5−16.4i)T2 1 + (1.73 - i)T + (9.5 - 16.4i)T^{2}
23 1+(−2+3.46i)T+(−11.5−19.9i)T2 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3+5.19i)T+(−14.5−25.1i)T2 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2}
31 1+4iT−31T2 1 + 4iT - 31T^{2}
37 1+(−9.52−5.5i)T+(18.5+32.0i)T2 1 + (-9.52 - 5.5i)T + (18.5 + 32.0i)T^{2}
41 1+(6.92+4i)T+(20.5+35.5i)T2 1 + (6.92 + 4i)T + (20.5 + 35.5i)T^{2}
43 1+(0.5+0.866i)T+(−21.5+37.2i)T2 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2}
47 1+9iT−47T2 1 + 9iT - 47T^{2}
53 1+12T+53T2 1 + 12T + 53T^{2}
59 1+(5.19−3i)T+(29.5−51.0i)T2 1 + (5.19 - 3i)T + (29.5 - 51.0i)T^{2}
61 1+(−30.5+52.8i)T2 1 + (-30.5 + 52.8i)T^{2}
67 1+(5.19+3i)T+(33.5+58.0i)T2 1 + (5.19 + 3i)T + (33.5 + 58.0i)T^{2}
71 1+(−6.06+3.5i)T+(35.5−61.4i)T2 1 + (-6.06 + 3.5i)T + (35.5 - 61.4i)T^{2}
73 1−2iT−73T2 1 - 2iT - 73T^{2}
79 1−12T+79T2 1 - 12T + 79T^{2}
83 1+16iT−83T2 1 + 16iT - 83T^{2}
89 1+(8.66+5i)T+(44.5+77.0i)T2 1 + (8.66 + 5i)T + (44.5 + 77.0i)T^{2}
97 1+(8.66−5i)T+(48.5−84.0i)T2 1 + (8.66 - 5i)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

−9.500472046488218777798812338431, −8.766173611373717048673526967359, −7.931905485928080433796206254554, −6.72974846728988621204162290515, −6.08462361058427509271749838180, −5.17760118262561966666870864745, −4.42579875972422036329490191242, −3.17015773683181319454697057457, −2.33340675723721922095429587519, −0.24680720501100630275765823600, 1.17869688948972168068245284548, 2.85385374284023703585571842619, 3.58903033016519027000351151858, 4.63167397604240893917344727222, 6.00528044251770672111656043103, 6.70025017608567996738905910853, 7.01667040069550958593972002565, 7.976690318795666859585653622294, 9.264591493723696602939310583613, 9.740893962891620478694304481628

Graph of the ZZ-function along the critical line