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Properties

Label 2-12-12.11-c7-0-5
Degree $2$
Conductor $12$
Sign $0.0829 + 0.996i$
Analytic cond. $3.74862$
Root an. cond. $1.93613$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.3 + 4.47i)2-s + (−31.1 + 34.8i)3-s + (88 − 92.9i)4-s − 205. i·5-s + (168. − 501. i)6-s − 999. i·7-s + (−498. + 1.35e3i)8-s + (−242. − 2.17e3i)9-s + (920 + 2.13e3i)10-s + 1.97e3·11-s + (496. + 5.96e3i)12-s − 1.27e4·13-s + (4.46e3 + 1.03e4i)14-s + (7.17e3 + 6.41e3i)15-s + (−896. − 1.63e4i)16-s − 1.88e4i·17-s + ⋯
L(s)  = 1  + (−0.918 + 0.395i)2-s + (−0.666 + 0.745i)3-s + (0.687 − 0.726i)4-s − 0.735i·5-s + (0.317 − 0.948i)6-s − 1.10i·7-s + (−0.344 + 0.938i)8-s + (−0.111 − 0.993i)9-s + (0.290 + 0.676i)10-s + 0.447·11-s + (0.0829 + 0.996i)12-s − 1.60·13-s + (0.435 + 1.01i)14-s + (0.548 + 0.490i)15-s + (−0.0546 − 0.998i)16-s − 0.930i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0829 + 0.996i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0829 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.0829 + 0.996i$
Analytic conductor: \(3.74862\)
Root analytic conductor: \(1.93613\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :7/2),\ 0.0829 + 0.996i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.347197 - 0.319504i\)
\(L(\frac12)\) \(\approx\) \(0.347197 - 0.319504i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (10.3 - 4.47i)T \)
3 \( 1 + (31.1 - 34.8i)T \)
good5 \( 1 + 205. iT - 7.81e4T^{2} \)
7 \( 1 + 999. iT - 8.23e5T^{2} \)
11 \( 1 - 1.97e3T + 1.94e7T^{2} \)
13 \( 1 + 1.27e4T + 6.27e7T^{2} \)
17 \( 1 + 1.88e4iT - 4.10e8T^{2} \)
19 \( 1 + 2.39e4iT - 8.93e8T^{2} \)
23 \( 1 + 7.23e4T + 3.40e9T^{2} \)
29 \( 1 - 1.06e5iT - 1.72e10T^{2} \)
31 \( 1 - 9.03e4iT - 2.75e10T^{2} \)
37 \( 1 - 4.33e4T + 9.49e10T^{2} \)
41 \( 1 + 7.85e5iT - 1.94e11T^{2} \)
43 \( 1 - 7.08e4iT - 2.71e11T^{2} \)
47 \( 1 + 5.63e5T + 5.06e11T^{2} \)
53 \( 1 - 3.48e5iT - 1.17e12T^{2} \)
59 \( 1 - 7.58e5T + 2.48e12T^{2} \)
61 \( 1 - 3.14e5T + 3.14e12T^{2} \)
67 \( 1 - 8.11e5iT - 6.06e12T^{2} \)
71 \( 1 + 2.65e5T + 9.09e12T^{2} \)
73 \( 1 + 2.59e5T + 1.10e13T^{2} \)
79 \( 1 + 5.23e6iT - 1.92e13T^{2} \)
83 \( 1 - 1.01e7T + 2.71e13T^{2} \)
89 \( 1 + 3.88e6iT - 4.42e13T^{2} \)
97 \( 1 - 7.24e6T + 8.07e13T^{2} \)
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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

−17.72329543583695631847866608590, −16.89480368139900730956985224805, −16.03246256813220137257544441649, −14.41649818967935099540945785497, −11.96906524196204569769380771503, −10.40744884795875416383243501229, −9.232252792017005568639828688244, −7.09175021848274678470365949358, −4.92190864043536808766193358771, −0.46445618493903922016043319982, 2.22195042962927585475181372723, 6.31400819264188877647603214996, 7.938236002908821722675550354989, 9.996807994387519671235422656562, 11.61152712056322090738830172542, 12.50541146134483754784196071845, 14.86222314403663166850411516370, 16.66067526651286276239280385004, 17.82281417238184685075871422963, 18.78635050264062439493460936320

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