Properties

Label 2-12-12.11-c11-0-12
Degree $2$
Conductor $12$
Sign $0.949 - 0.312i$
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 + (1.71e4 − 8.22e3i)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 + (8.18e5 − 2.69e5i)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 + (0.902 − 0.431i)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 + (0.949 − 0.312i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.949 - 0.312i$
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: \((2,\ 12,\ (\ :11/2),\ 0.949 - 0.312i)\)

Particular Values

\(L(6)\) \(\approx\) \(3.87953 + 0.621748i\)
\(L(\frac12)\) \(\approx\) \(3.87953 + 0.621748i\)
\(L(\frac{13}{2})\) 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

−17.63195023825653205579989904006, −15.47041089648220338864060316452, −14.33720618314680634074002211749, −13.77586770633099182659910568766, −11.89495422110388332244568319861, −10.30765920931882259265690848614, −7.46001726518005520430186019956, −6.64232775418364914285596803205, −3.68014673903788323729504833592, −2.37320684693563375402999678851, 1.88056380268556190092965814231, 4.08057500204924060580137561668, 5.38443518454005791519367248434, 8.228871898903973847729470185142, 9.689654350317624694751198454685, 12.03070897229269813180899973174, 13.04954048621681648300138715318, 14.49994084469257062017170752489, 15.79370636040615156474001498624, 16.75362060064551202608603531043

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