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Properties

Label 2-12-4.3-c16-0-3
Degree $2$
Conductor $12$
Sign $-0.983 + 0.178i$
Analytic cond. $19.4789$
Root an. cond. $4.41349$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (196. + 164. i)2-s + 3.78e3i·3-s + (1.16e4 + 6.44e4i)4-s − 4.50e4·5-s + (−6.21e5 + 7.44e5i)6-s + 3.24e6i·7-s + (−8.28e6 + 1.45e7i)8-s − 1.43e7·9-s + (−8.84e6 − 7.38e6i)10-s + 2.75e7i·11-s + (−2.44e8 + 4.42e7i)12-s − 6.92e8·13-s + (−5.31e8 + 6.36e8i)14-s − 1.70e8i·15-s + (−4.02e9 + 1.50e9i)16-s + 2.60e9·17-s + ⋯
L(s)  = 1  + (0.767 + 0.640i)2-s + 0.577i·3-s + (0.178 + 0.983i)4-s − 0.115·5-s + (−0.370 + 0.443i)6-s + 0.562i·7-s + (−0.493 + 0.869i)8-s − 0.333·9-s + (−0.0884 − 0.0738i)10-s + 0.128i·11-s + (−0.568 + 0.102i)12-s − 0.848·13-s + (−0.360 + 0.431i)14-s − 0.0665i·15-s + (−0.936 + 0.351i)16-s + 0.373·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.983 + 0.178i$
Analytic conductor: \(19.4789\)
Root analytic conductor: \(4.41349\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :8),\ -0.983 + 0.178i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.172696 - 1.92095i\)
\(L(\frac12)\) \(\approx\) \(0.172696 - 1.92095i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-196. - 164. i)T \)
3 \( 1 - 3.78e3iT \)
good5 \( 1 + 4.50e4T + 1.52e11T^{2} \)
7 \( 1 - 3.24e6iT - 3.32e13T^{2} \)
11 \( 1 - 2.75e7iT - 4.59e16T^{2} \)
13 \( 1 + 6.92e8T + 6.65e17T^{2} \)
17 \( 1 - 2.60e9T + 4.86e19T^{2} \)
19 \( 1 + 1.17e10iT - 2.88e20T^{2} \)
23 \( 1 - 3.87e9iT - 6.13e21T^{2} \)
29 \( 1 + 6.46e11T + 2.50e23T^{2} \)
31 \( 1 - 9.55e11iT - 7.27e23T^{2} \)
37 \( 1 - 6.34e12T + 1.23e25T^{2} \)
41 \( 1 - 1.16e13T + 6.37e25T^{2} \)
43 \( 1 - 1.79e13iT - 1.36e26T^{2} \)
47 \( 1 - 3.29e13iT - 5.66e26T^{2} \)
53 \( 1 - 5.59e12T + 3.87e27T^{2} \)
59 \( 1 - 1.93e14iT - 2.15e28T^{2} \)
61 \( 1 + 1.41e13T + 3.67e28T^{2} \)
67 \( 1 + 3.51e14iT - 1.64e29T^{2} \)
71 \( 1 - 8.84e14iT - 4.16e29T^{2} \)
73 \( 1 - 1.70e14T + 6.50e29T^{2} \)
79 \( 1 + 1.50e15iT - 2.30e30T^{2} \)
83 \( 1 + 4.88e14iT - 5.07e30T^{2} \)
89 \( 1 + 7.27e15T + 1.54e31T^{2} \)
97 \( 1 + 1.59e15T + 6.14e31T^{2} \)
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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

−16.52688913057725218831500616312, −15.37791319783663153395746198219, −14.40590770990751541047002854165, −12.75973850661548613329021911265, −11.41652151363990823500754799674, −9.324581675754234548872603233980, −7.62773763194188765134692585247, −5.79776346750630117402755542169, −4.42089597385668137827988435624, −2.72852938485569419770076332942, 0.52316511956028171752043080997, 2.15956597324740932130136324824, 3.92601041506002522614583317643, 5.75450975967212034497768258007, 7.47417069474909825215345524244, 9.795956943519411247359031310279, 11.34618845947562767203137039191, 12.58931665460856507779963763412, 13.78617438618749897722157911645, 14.97227782416766960413942158245

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