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Properties

Label 2-640-80.13-c2-0-27
Degree $2$
Conductor $640$
Sign $0.182 + 0.983i$
Analytic cond. $17.4387$
Root an. cond. $4.17597$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.83i·3-s + (−0.146 + 4.99i)5-s + (−1.69 − 1.69i)7-s − 5.67·9-s + (6.09 − 6.09i)11-s + 14.9i·13-s + (19.1 + 0.563i)15-s + (10.0 − 10.0i)17-s + (−4.15 + 4.15i)19-s + (−6.50 + 6.50i)21-s + (31.1 − 31.1i)23-s + (−24.9 − 1.46i)25-s − 12.7i·27-s + (38.9 − 38.9i)29-s − 15.2·31-s + ⋯
L(s)  = 1  − 1.27i·3-s + (−0.0293 + 0.999i)5-s + (−0.242 − 0.242i)7-s − 0.630·9-s + (0.554 − 0.554i)11-s + 1.14i·13-s + (1.27 + 0.0375i)15-s + (0.589 − 0.589i)17-s + (−0.218 + 0.218i)19-s + (−0.309 + 0.309i)21-s + (1.35 − 1.35i)23-s + (−0.998 − 0.0587i)25-s − 0.471i·27-s + (1.34 − 1.34i)29-s − 0.492·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.182 + 0.983i$
Analytic conductor: \(17.4387\)
Root analytic conductor: \(4.17597\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1),\ 0.182 + 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.774234715\)
\(L(\frac12)\) \(\approx\) \(1.774234715\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.146 - 4.99i)T \)
good3 \( 1 + 3.83iT - 9T^{2} \)
7 \( 1 + (1.69 + 1.69i)T + 49iT^{2} \)
11 \( 1 + (-6.09 + 6.09i)T - 121iT^{2} \)
13 \( 1 - 14.9iT - 169T^{2} \)
17 \( 1 + (-10.0 + 10.0i)T - 289iT^{2} \)
19 \( 1 + (4.15 - 4.15i)T - 361iT^{2} \)
23 \( 1 + (-31.1 + 31.1i)T - 529iT^{2} \)
29 \( 1 + (-38.9 + 38.9i)T - 841iT^{2} \)
31 \( 1 + 15.2T + 961T^{2} \)
37 \( 1 - 10.0iT - 1.36e3T^{2} \)
41 \( 1 - 17.1iT - 1.68e3T^{2} \)
43 \( 1 - 41.2T + 1.84e3T^{2} \)
47 \( 1 + (35.1 - 35.1i)T - 2.20e3iT^{2} \)
53 \( 1 - 5.40T + 2.80e3T^{2} \)
59 \( 1 + (13.6 + 13.6i)T + 3.48e3iT^{2} \)
61 \( 1 + (55.0 + 55.0i)T + 3.72e3iT^{2} \)
67 \( 1 - 67.6T + 4.48e3T^{2} \)
71 \( 1 - 68.8iT - 5.04e3T^{2} \)
73 \( 1 + (-84.8 + 84.8i)T - 5.32e3iT^{2} \)
79 \( 1 + 89.5iT - 6.24e3T^{2} \)
83 \( 1 + 128. iT - 6.88e3T^{2} \)
89 \( 1 + 43.6T + 7.92e3T^{2} \)
97 \( 1 + (-50.0 + 50.0i)T - 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.25187076966135319963480207202, −9.279099792162920588453205250503, −8.207117601488685153112386339290, −7.33907983518698867114447094661, −6.56394605313993738351654556130, −6.24553615724405450703678585203, −4.52087964337221088387116349086, −3.21020078736524097545079788105, −2.16446212804938334183254165228, −0.76917367433113559721910235522, 1.22775155350437235133535067143, 3.14826092119661162987419953889, 4.06408356863286738708461156048, 5.06942626325743625587003672361, 5.59379538517647958080756431867, 7.05846104447874903692591103887, 8.210459934066373058353488697645, 9.087532053745725556711072029621, 9.583618610871133077243095713153, 10.43970184714680470434791039534

Graph of the $Z$-function along the critical line