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Properties

Label 2-3248-203.6-c0-0-0
Degree 22
Conductor 32483248
Sign 0.357−0.934i0.357 - 0.934i
Analytic cond. 1.620961.62096
Root an. cond. 1.273171.27317
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  + (−0.222 + 0.974i)7-s + (0.900 + 0.433i)9-s + (0.376 + 0.781i)11-s + (−0.277 − 0.347i)23-s + (−0.222 − 0.974i)25-s + (0.222 + 0.974i)29-s + (−0.678 + 1.40i)37-s + (−0.900 − 0.433i)49-s + (−1.12 + 1.40i)53-s + (−0.623 + 0.781i)63-s + (0.400 + 0.193i)67-s + (1.12 − 0.541i)71-s + (−0.846 + 0.193i)77-s + (0.376 − 0.781i)79-s + (0.623 + 0.781i)81-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)7-s + (0.900 + 0.433i)9-s + (0.376 + 0.781i)11-s + (−0.277 − 0.347i)23-s + (−0.222 − 0.974i)25-s + (0.222 + 0.974i)29-s + (−0.678 + 1.40i)37-s + (−0.900 − 0.433i)49-s + (−1.12 + 1.40i)53-s + (−0.623 + 0.781i)63-s + (0.400 + 0.193i)67-s + (1.12 − 0.541i)71-s + (−0.846 + 0.193i)77-s + (0.376 − 0.781i)79-s + (0.623 + 0.781i)81-s + ⋯

Functional equation

Λ(s)=(3248s/2 ΓC(s) L(s)=((0.357−0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3248s/2 ΓC(s) L(s)=((0.357−0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32483248    =    24⋅7⋅292^{4} \cdot 7 \cdot 29
Sign: 0.357−0.934i0.357 - 0.934i
Analytic conductor: 1.620961.62096
Root analytic conductor: 1.273171.27317
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3248(209,⋅)\chi_{3248} (209, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3248, ( :0), 0.357−0.934i)(2,\ 3248,\ (\ :0),\ 0.357 - 0.934i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2642555681.264255568
L(12)L(\frac12) ≈\approx 1.2642555681.264255568
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 1
7 1+(0.222−0.974i)T 1 + (0.222 - 0.974i)T
29 1+(−0.222−0.974i)T 1 + (-0.222 - 0.974i)T
good3 1+(−0.900−0.433i)T2 1 + (-0.900 - 0.433i)T^{2}
5 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
11 1+(−0.376−0.781i)T+(−0.623+0.781i)T2 1 + (-0.376 - 0.781i)T + (-0.623 + 0.781i)T^{2}
13 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
17 1+T2 1 + T^{2}
19 1+(−0.900+0.433i)T2 1 + (-0.900 + 0.433i)T^{2}
23 1+(0.277+0.347i)T+(−0.222+0.974i)T2 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2}
31 1+(−0.222−0.974i)T2 1 + (-0.222 - 0.974i)T^{2}
37 1+(0.678−1.40i)T+(−0.623−0.781i)T2 1 + (0.678 - 1.40i)T + (-0.623 - 0.781i)T^{2}
41 1+T2 1 + T^{2}
43 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
47 1+(0.623−0.781i)T2 1 + (0.623 - 0.781i)T^{2}
53 1+(1.12−1.40i)T+(−0.222−0.974i)T2 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2}
59 1−T2 1 - T^{2}
61 1+(−0.900−0.433i)T2 1 + (-0.900 - 0.433i)T^{2}
67 1+(−0.400−0.193i)T+(0.623+0.781i)T2 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2}
71 1+(−1.12+0.541i)T+(0.623−0.781i)T2 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2}
73 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
79 1+(−0.376+0.781i)T+(−0.623−0.781i)T2 1 + (-0.376 + 0.781i)T + (-0.623 - 0.781i)T^{2}
83 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
89 1+(−0.222−0.974i)T2 1 + (-0.222 - 0.974i)T^{2}
97 1+(−0.900+0.433i)T2 1 + (-0.900 + 0.433i)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

−8.958535430600292714106593189108, −8.282217675240105798364042325683, −7.47363590818375023161561124943, −6.68290604469608675209232229196, −6.09369047018674948919810364532, −4.98122036037618858423559278721, −4.52732743907697331027155568157, −3.41692099026906464755656954287, −2.37750061089451862226181574377, −1.54285825530847634013732895610, 0.808949316077380735617971902848, 1.92882671218186995726524193540, 3.44338176199247014893268001369, 3.81171059014286633142959704676, 4.73662916046925183358453560516, 5.76038718945470313574510302159, 6.53982792378210888558529631643, 7.18520896450438175698033609340, 7.83292294740137829861480640739, 8.699617340441510275370001436011

Graph of the ZZ-function along the critical line