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Properties

Label 2-12-12.11-c11-0-1
Degree $2$
Conductor $12$
Sign $-0.673 + 0.739i$
Analytic cond. $9.22011$
Root an. cond. $3.03646$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−44.8 + 5.79i)2-s + (−353. + 228. i)3-s + (1.98e3 − 520. i)4-s + 1.26e4i·5-s + (1.45e4 − 1.23e4i)6-s + 3.21e4i·7-s + (−8.58e4 + 3.48e4i)8-s + (7.24e4 − 1.61e5i)9-s + (−7.33e4 − 5.67e5i)10-s − 2.50e5·11-s + (−5.80e5 + 6.37e5i)12-s − 9.76e5·13-s + (−1.86e5 − 1.44e6i)14-s + (−2.89e6 − 4.46e6i)15-s + (3.65e6 − 2.06e6i)16-s − 2.81e6i·17-s + ⋯
L(s)  = 1  + (−0.991 + 0.128i)2-s + (−0.839 + 0.543i)3-s + (0.967 − 0.254i)4-s + 1.81i·5-s + (0.762 − 0.646i)6-s + 0.723i·7-s + (−0.926 + 0.376i)8-s + (0.408 − 0.912i)9-s + (−0.232 − 1.79i)10-s − 0.469·11-s + (−0.673 + 0.739i)12-s − 0.729·13-s + (−0.0927 − 0.717i)14-s + (−0.984 − 1.51i)15-s + (0.870 − 0.491i)16-s − 0.481i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.673 + 0.739i$
Analytic conductor: \(9.22011\)
Root analytic conductor: \(3.03646\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :11/2),\ -0.673 + 0.739i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.126081 - 0.285467i\)
\(L(\frac12)\) \(\approx\) \(0.126081 - 0.285467i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (44.8 - 5.79i)T \)
3 \( 1 + (353. - 228. i)T \)
good5 \( 1 - 1.26e4iT - 4.88e7T^{2} \)
7 \( 1 - 3.21e4iT - 1.97e9T^{2} \)
11 \( 1 + 2.50e5T + 2.85e11T^{2} \)
13 \( 1 + 9.76e5T + 1.79e12T^{2} \)
17 \( 1 + 2.81e6iT - 3.42e13T^{2} \)
19 \( 1 - 6.17e6iT - 1.16e14T^{2} \)
23 \( 1 - 3.69e7T + 9.52e14T^{2} \)
29 \( 1 + 9.70e7iT - 1.22e16T^{2} \)
31 \( 1 + 2.18e8iT - 2.54e16T^{2} \)
37 \( 1 + 4.00e8T + 1.77e17T^{2} \)
41 \( 1 - 3.38e7iT - 5.50e17T^{2} \)
43 \( 1 - 9.95e8iT - 9.29e17T^{2} \)
47 \( 1 - 4.43e8T + 2.47e18T^{2} \)
53 \( 1 - 3.45e9iT - 9.26e18T^{2} \)
59 \( 1 + 5.47e9T + 3.01e19T^{2} \)
61 \( 1 - 2.88e9T + 4.35e19T^{2} \)
67 \( 1 - 1.87e9iT - 1.22e20T^{2} \)
71 \( 1 + 2.17e10T + 2.31e20T^{2} \)
73 \( 1 + 1.38e10T + 3.13e20T^{2} \)
79 \( 1 - 2.31e10iT - 7.47e20T^{2} \)
83 \( 1 + 1.51e10T + 1.28e21T^{2} \)
89 \( 1 - 3.51e10iT - 2.77e21T^{2} \)
97 \( 1 - 3.62e10T + 7.15e21T^{2} \)
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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

−18.32698441919690707841048396960, −17.17637103013241505558267742557, −15.56305249665408764594433012773, −14.81523541245498178008417785617, −11.80419704963465802239288672376, −10.75828167877711390529118172394, −9.675917057478243334249233069557, −7.29042193552724891729608455589, −5.93110905948087315254961082782, −2.72682960924347582145303729471, 0.24001635838488139600619653385, 1.40239413463535982488294762396, 5.09929778160003321615406663944, 7.23894779873118168791852027766, 8.754723263650746855656587528656, 10.50029346379849618880520431966, 12.12122527376990798658124873015, 13.06672134275930259852936745002, 15.95668477046943729807888686171, 16.93864564470712865406674395727

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