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Properties

Label 2-12-4.3-c20-0-4
Degree $2$
Conductor $12$
Sign $0.330 - 0.943i$
Analytic cond. $30.4216$
Root an. cond. $5.51558$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (171. − 1.00e3i)2-s + 3.40e4i·3-s + (−9.89e5 − 3.46e5i)4-s + 1.74e7·5-s + (3.44e7 + 5.85e6i)6-s + 3.10e8i·7-s + (−5.19e8 + 9.39e8i)8-s − 1.16e9·9-s + (2.99e9 − 1.76e10i)10-s + 2.01e9i·11-s + (1.18e10 − 3.37e10i)12-s − 1.45e11·13-s + (3.13e11 + 5.33e10i)14-s + 5.94e11i·15-s + (8.59e11 + 6.86e11i)16-s − 3.62e12·17-s + ⋯
L(s)  = 1  + (0.167 − 0.985i)2-s + 0.577i·3-s + (−0.943 − 0.330i)4-s + 1.78·5-s + (0.569 + 0.0968i)6-s + 1.10i·7-s + (−0.484 + 0.874i)8-s − 0.333·9-s + (0.299 − 1.76i)10-s + 0.0776i·11-s + (0.190 − 0.544i)12-s − 1.05·13-s + (1.08 + 0.184i)14-s + 1.03i·15-s + (0.781 + 0.624i)16-s − 1.80·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.330 - 0.943i$
Analytic conductor: \(30.4216\)
Root analytic conductor: \(5.51558\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :10),\ 0.330 - 0.943i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(1.688474656\)
\(L(\frac12)\) \(\approx\) \(1.688474656\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-171. + 1.00e3i)T \)
3 \( 1 - 3.40e4iT \)
good5 \( 1 - 1.74e7T + 9.53e13T^{2} \)
7 \( 1 - 3.10e8iT - 7.97e16T^{2} \)
11 \( 1 - 2.01e9iT - 6.72e20T^{2} \)
13 \( 1 + 1.45e11T + 1.90e22T^{2} \)
17 \( 1 + 3.62e12T + 4.06e24T^{2} \)
19 \( 1 + 3.83e11iT - 3.75e25T^{2} \)
23 \( 1 - 7.19e13iT - 1.71e27T^{2} \)
29 \( 1 + 2.72e14T + 1.76e29T^{2} \)
31 \( 1 - 3.17e14iT - 6.71e29T^{2} \)
37 \( 1 - 1.82e15T + 2.31e31T^{2} \)
41 \( 1 - 7.50e15T + 1.80e32T^{2} \)
43 \( 1 - 9.17e15iT - 4.67e32T^{2} \)
47 \( 1 - 7.76e16iT - 2.76e33T^{2} \)
53 \( 1 + 5.62e16T + 3.05e34T^{2} \)
59 \( 1 + 3.58e17iT - 2.61e35T^{2} \)
61 \( 1 + 6.85e17T + 5.08e35T^{2} \)
67 \( 1 - 1.47e18iT - 3.32e36T^{2} \)
71 \( 1 + 5.13e18iT - 1.05e37T^{2} \)
73 \( 1 - 8.06e17T + 1.84e37T^{2} \)
79 \( 1 - 1.03e19iT - 8.96e37T^{2} \)
83 \( 1 - 2.51e19iT - 2.40e38T^{2} \)
89 \( 1 + 2.22e18T + 9.72e38T^{2} \)
97 \( 1 - 5.14e19T + 5.43e39T^{2} \)
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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

−15.18039416499978401178303235817, −13.89675786506340656494360295719, −12.74223873563335195184343115354, −11.12651507631305010175797190181, −9.648918855250819221512928804090, −9.126364629273776941157112887367, −5.90195331073157242511498752816, −4.86902870138973751942562124107, −2.69260237241838797429041187428, −1.83602366771352615704004110225, 0.46136692795183301335723454395, 2.25396181081471286419438952962, 4.66273491035336726005534266111, 6.18542695030811600606496966432, 7.12958420967936732271155495270, 8.966491440808276245205118487835, 10.31620986670197177809108828278, 12.91404734224499768391121158937, 13.65648097040950141576165051990, 14.63175056565729632689299229747

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