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Properties

Label 2-6-3.2-c10-0-2
Degree $2$
Conductor $6$
Sign $0.0775 + 0.996i$
Analytic cond. $3.81214$
Root an. cond. $1.95247$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22.6i·2-s + (242. − 18.8i)3-s − 512.·4-s − 4.81e3i·5-s + (−426. − 5.48e3i)6-s + 670.·7-s + 1.15e4i·8-s + (5.83e4 − 9.12e3i)9-s − 1.09e5·10-s + 2.33e5i·11-s + (−1.24e5 + 9.64e3i)12-s + 3.07e5·13-s − 1.51e4i·14-s + (−9.07e4 − 1.16e6i)15-s + 2.62e5·16-s + 6.72e5i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.996 − 0.0775i)3-s − 0.500·4-s − 1.54i·5-s + (−0.0548 − 0.704i)6-s + 0.0398·7-s + 0.353i·8-s + (0.987 − 0.154i)9-s − 1.09·10-s + 1.44i·11-s + (−0.498 + 0.0387i)12-s + 0.828·13-s − 0.0282i·14-s + (−0.119 − 1.53i)15-s + 0.250·16-s + 0.473i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0775 + 0.996i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.0775 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.0775 + 0.996i$
Analytic conductor: \(3.81214\)
Root analytic conductor: \(1.95247\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :5),\ 0.0775 + 0.996i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.35398 - 1.25281i\)
\(L(\frac12)\) \(\approx\) \(1.35398 - 1.25281i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 22.6iT \)
3 \( 1 + (-242. + 18.8i)T \)
good5 \( 1 + 4.81e3iT - 9.76e6T^{2} \)
7 \( 1 - 670.T + 2.82e8T^{2} \)
11 \( 1 - 2.33e5iT - 2.59e10T^{2} \)
13 \( 1 - 3.07e5T + 1.37e11T^{2} \)
17 \( 1 - 6.72e5iT - 2.01e12T^{2} \)
19 \( 1 + 1.55e6T + 6.13e12T^{2} \)
23 \( 1 - 5.57e6iT - 4.14e13T^{2} \)
29 \( 1 + 2.97e7iT - 4.20e14T^{2} \)
31 \( 1 - 3.09e7T + 8.19e14T^{2} \)
37 \( 1 + 8.56e7T + 4.80e15T^{2} \)
41 \( 1 - 3.59e7iT - 1.34e16T^{2} \)
43 \( 1 + 3.66e7T + 2.16e16T^{2} \)
47 \( 1 - 3.28e7iT - 5.25e16T^{2} \)
53 \( 1 - 4.59e8iT - 1.74e17T^{2} \)
59 \( 1 + 4.88e8iT - 5.11e17T^{2} \)
61 \( 1 + 6.12e7T + 7.13e17T^{2} \)
67 \( 1 + 6.70e8T + 1.82e18T^{2} \)
71 \( 1 - 1.23e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.08e9T + 4.29e18T^{2} \)
79 \( 1 + 1.86e9T + 9.46e18T^{2} \)
83 \( 1 + 1.09e9iT - 1.55e19T^{2} \)
89 \( 1 - 5.19e9iT - 3.11e19T^{2} \)
97 \( 1 + 1.07e10T + 7.37e19T^{2} \)
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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

−20.47236383693974337266432558219, −19.34299920599527337211571847162, −17.45639467293969378016998032104, −15.45874697311626271191912205813, −13.44648530900008741847180894762, −12.37259579454103471612235173481, −9.681806365547529709097802400295, −8.314346235040417047998142046242, −4.34993008899755257657202656228, −1.60206598405182890542098816439, 3.25733308796666326014871595297, 6.67324175281569157318129879835, 8.494906272449475173116398799368, 10.64154244021490082900169321477, 13.68786564713015031078865121610, 14.63191173228698903395864232520, 16.01220147851931185966576554706, 18.32292530179187238173311580401, 19.15973450640303357086177402926, 21.26636118011910904786803904190

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