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Properties

Label 2-12-4.3-c16-0-4
Degree $2$
Conductor $12$
Sign $0.825 + 0.564i$
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  + (−119. − 226. i)2-s − 3.78e3i·3-s + (−3.69e4 + 5.41e4i)4-s − 4.81e5·5-s + (−8.57e5 + 4.52e5i)6-s + 5.15e5i·7-s + (1.66e7 + 1.91e6i)8-s − 1.43e7·9-s + (5.75e7 + 1.09e8i)10-s + 1.66e8i·11-s + (2.04e8 + 1.40e8i)12-s − 4.92e8·13-s + (1.16e8 − 6.15e7i)14-s + 1.82e9i·15-s + (−1.55e9 − 4.00e9i)16-s + 7.46e9·17-s + ⋯
L(s)  = 1  + (−0.466 − 0.884i)2-s − 0.577i·3-s + (−0.564 + 0.825i)4-s − 1.23·5-s + (−0.510 + 0.269i)6-s + 0.0893i·7-s + (0.993 + 0.113i)8-s − 0.333·9-s + (0.575 + 1.09i)10-s + 0.775i·11-s + (0.476 + 0.325i)12-s − 0.604·13-s + (0.0790 − 0.0417i)14-s + 0.711i·15-s + (−0.362 − 0.931i)16-s + 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.825 + 0.564i$
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: \((2,\ 12,\ (\ :8),\ 0.825 + 0.564i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.820009 - 0.253507i\)
\(L(\frac12)\) \(\approx\) \(0.820009 - 0.253507i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (119. + 226. i)T \)
3 \( 1 + 3.78e3iT \)
good5 \( 1 + 4.81e5T + 1.52e11T^{2} \)
7 \( 1 - 5.15e5iT - 3.32e13T^{2} \)
11 \( 1 - 1.66e8iT - 4.59e16T^{2} \)
13 \( 1 + 4.92e8T + 6.65e17T^{2} \)
17 \( 1 - 7.46e9T + 4.86e19T^{2} \)
19 \( 1 - 3.55e9iT - 2.88e20T^{2} \)
23 \( 1 + 1.41e11iT - 6.13e21T^{2} \)
29 \( 1 - 9.07e11T + 2.50e23T^{2} \)
31 \( 1 - 1.71e11iT - 7.27e23T^{2} \)
37 \( 1 + 3.92e12T + 1.23e25T^{2} \)
41 \( 1 - 6.33e12T + 6.37e25T^{2} \)
43 \( 1 - 1.38e13iT - 1.36e26T^{2} \)
47 \( 1 - 3.46e13iT - 5.66e26T^{2} \)
53 \( 1 - 8.06e13T + 3.87e27T^{2} \)
59 \( 1 - 1.70e14iT - 2.15e28T^{2} \)
61 \( 1 - 3.04e14T + 3.67e28T^{2} \)
67 \( 1 - 3.16e14iT - 1.64e29T^{2} \)
71 \( 1 + 1.80e14iT - 4.16e29T^{2} \)
73 \( 1 + 1.28e15T + 6.50e29T^{2} \)
79 \( 1 + 1.91e15iT - 2.30e30T^{2} \)
83 \( 1 - 2.98e15iT - 5.07e30T^{2} \)
89 \( 1 + 8.50e14T + 1.54e31T^{2} \)
97 \( 1 + 1.30e15T + 6.14e31T^{2} \)
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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

−16.28341307146337277143386580377, −14.46148618451809886038046791056, −12.50981379880280013518411300275, −11.93732678338092189875995435371, −10.29244447115550912972938471918, −8.427267393218790076062421787486, −7.30303638010695352938263453658, −4.40881827306690316217908575669, −2.70727303487181833561538748145, −0.855987303459895895914255798529, 0.55950069651480980350766947969, 3.72605900442154651581777272146, 5.34906942686574631778549457582, 7.34496955963939689331477414868, 8.562075554285211680753851591548, 10.16824305542066946533075318283, 11.73400865982265849522249723391, 13.97273183595459424130520722246, 15.31758758362267838735934569132, 16.10537237234048082030069025234

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