Properties

Label 2-6-3.2-c18-0-3
Degree $2$
Conductor $6$
Sign $0.257 + 0.966i$
Analytic cond. $12.3231$
Root an. cond. $3.51043$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 362. i·2-s + (−1.90e4 + 5.06e3i)3-s − 1.31e5·4-s + 2.98e6i·5-s + (−1.83e6 − 6.88e6i)6-s − 4.92e7·7-s − 4.74e7i·8-s + (3.36e8 − 1.92e8i)9-s − 1.08e9·10-s − 5.85e8i·11-s + (2.49e9 − 6.64e8i)12-s + 1.21e10·13-s − 1.78e10i·14-s + (−1.51e10 − 5.68e10i)15-s + 1.71e10·16-s − 1.49e11i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.966 + 0.257i)3-s − 0.500·4-s + 1.53i·5-s + (−0.182 − 0.683i)6-s − 1.22·7-s − 0.353i·8-s + (0.867 − 0.497i)9-s − 1.08·10-s − 0.248i·11-s + (0.483 − 0.128i)12-s + 1.14·13-s − 0.863i·14-s + (−0.394 − 1.47i)15-s + 0.250·16-s − 1.26i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.257 + 0.966i$
Analytic conductor: \(12.3231\)
Root analytic conductor: \(3.51043\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :9),\ 0.257 + 0.966i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.00457753 - 0.00351748i\)
\(L(\frac12)\) \(\approx\) \(0.00457753 - 0.00351748i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 362. iT \)
3 \( 1 + (1.90e4 - 5.06e3i)T \)
good5 \( 1 - 2.98e6iT - 3.81e12T^{2} \)
7 \( 1 + 4.92e7T + 1.62e15T^{2} \)
11 \( 1 + 5.85e8iT - 5.55e18T^{2} \)
13 \( 1 - 1.21e10T + 1.12e20T^{2} \)
17 \( 1 + 1.49e11iT - 1.40e22T^{2} \)
19 \( 1 + 3.13e11T + 1.04e23T^{2} \)
23 \( 1 - 2.41e12iT - 3.24e24T^{2} \)
29 \( 1 + 1.39e13iT - 2.10e26T^{2} \)
31 \( 1 + 2.16e13T + 6.99e26T^{2} \)
37 \( 1 - 2.89e13T + 1.68e28T^{2} \)
41 \( 1 - 2.60e14iT - 1.07e29T^{2} \)
43 \( 1 + 9.12e13T + 2.52e29T^{2} \)
47 \( 1 + 1.34e15iT - 1.25e30T^{2} \)
53 \( 1 + 4.92e15iT - 1.08e31T^{2} \)
59 \( 1 + 1.63e15iT - 7.50e31T^{2} \)
61 \( 1 + 1.27e16T + 1.36e32T^{2} \)
67 \( 1 - 1.62e16T + 7.40e32T^{2} \)
71 \( 1 + 3.19e16iT - 2.10e33T^{2} \)
73 \( 1 + 6.00e16T + 3.46e33T^{2} \)
79 \( 1 + 1.80e17T + 1.43e34T^{2} \)
83 \( 1 - 8.49e16iT - 3.49e34T^{2} \)
89 \( 1 - 5.84e17iT - 1.22e35T^{2} \)
97 \( 1 + 1.70e17T + 5.77e35T^{2} \)
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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

−18.00726760197210690296629671897, −16.34037577570337190945402210503, −15.28002393663555026796355039246, −13.41706102605803721078177204396, −11.24425454621489668856055647558, −9.793862659824189282330190637081, −6.95601463872881762348848010645, −6.00557962546592460698305099099, −3.51252611718057511956965275021, −0.00306737089505830738251974423, 1.36447798711501239219946032905, 4.24984511827496006736782550584, 6.04050552282655351767962832158, 8.808047610760058026605869492421, 10.57751384456839864342271764464, 12.56126893844413146007477848754, 12.93698323774024478049552779189, 16.11054790008726661032457360954, 17.09706261204694073565059913925, 18.81165512256499529327187625592

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