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Properties

Label 2-3240-360.149-c0-0-5
Degree 22
Conductor 32403240
Sign 0.342−0.939i0.342 - 0.939i
Analytic cond. 1.616971.61697
Root an. cond. 1.271601.27160
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999i·8-s + (−0.499 − 0.866i)10-s + (−0.5 + 0.866i)16-s + 1.73·17-s + 1.73i·19-s − 0.999i·20-s + (0.866 − 1.5i)23-s + (0.499 + 0.866i)25-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.499i)32-s + (1.49 + 0.866i)34-s + (−0.866 + 1.49i)38-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999i·8-s + (−0.499 − 0.866i)10-s + (−0.5 + 0.866i)16-s + 1.73·17-s + 1.73i·19-s − 0.999i·20-s + (0.866 − 1.5i)23-s + (0.499 + 0.866i)25-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.499i)32-s + (1.49 + 0.866i)34-s + (−0.866 + 1.49i)38-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((0.342−0.939i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s) L(s)=((0.342−0.939i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 0.342−0.939i0.342 - 0.939i
Analytic conductor: 1.616971.61697
Root analytic conductor: 1.271601.27160
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3240(1349,⋅)\chi_{3240} (1349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :0), 0.342−0.939i)(2,\ 3240,\ (\ :0),\ 0.342 - 0.939i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8656028911.865602891
L(12)L(\frac12) ≈\approx 1.8656028911.865602891
L(1)L(1) 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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
3 1 1
5 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
good7 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
13 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
17 1−1.73T+T2 1 - 1.73T + T^{2}
19 1−1.73iT−T2 1 - 1.73iT - T^{2}
23 1+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
29 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
31 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+T2 1 + T^{2}
41 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
43 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
47 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
53 1−iT−T2 1 - iT - T^{2}
59 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
61 1+(−1.5+0.866i)T+(0.5−0.866i)T2 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2}
67 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
83 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)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

−8.553878682989301208468220561294, −8.146273068801059516560121516847, −7.48306610621289982435207642923, −6.75168984452561472778122541531, −5.78320205913641770868145414206, −5.20105013387499800074080846780, −4.34320634627157855201372412701, −3.61666391580642632152600668482, −2.91420861413574966714953470237, −1.39946721765067327501956454241, 0.966762182452367038959833914728, 2.38658944902249868862699106629, 3.30861555030007517803080849281, 3.74629247742254120842019771332, 4.86588434927753064853923439152, 5.40736776190717007796477326596, 6.39262453117356248693827462754, 7.24428023627683891989971176666, 7.59728141348824773390490575469, 8.759346748622231336065147496119

Graph of the ZZ-function along the critical line