Properties

Label 2-12-4.3-c10-0-8
Degree $2$
Conductor $12$
Sign $-0.984 + 0.175i$
Analytic cond. $7.62428$
Root an. cond. $2.76121$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (20.5 − 24.5i)2-s + 140. i·3-s + (−179. − 1.00e3i)4-s − 4.87e3·5-s + (3.44e3 + 2.88e3i)6-s − 2.26e4i·7-s + (−2.84e4 − 1.63e4i)8-s − 1.96e4·9-s + (−1.00e5 + 1.19e5i)10-s + 7.73e4i·11-s + (1.41e5 − 2.51e4i)12-s − 3.19e5·13-s + (−5.55e5 − 4.65e5i)14-s − 6.83e5i·15-s + (−9.84e5 + 3.61e5i)16-s + 2.42e6·17-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)2-s + 0.577i·3-s + (−0.175 − 0.984i)4-s − 1.55·5-s + (0.442 + 0.370i)6-s − 1.34i·7-s + (−0.867 − 0.498i)8-s − 0.333·9-s + (−1.00 + 1.19i)10-s + 0.480i·11-s + (0.568 − 0.101i)12-s − 0.861·13-s + (−1.03 − 0.864i)14-s − 0.900i·15-s + (−0.938 + 0.344i)16-s + 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.984 + 0.175i$
Analytic conductor: \(7.62428\)
Root analytic conductor: \(2.76121\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :5),\ -0.984 + 0.175i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0847882 - 0.960689i\)
\(L(\frac12)\) \(\approx\) \(0.0847882 - 0.960689i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-20.5 + 24.5i)T \)
3 \( 1 - 140. iT \)
good5 \( 1 + 4.87e3T + 9.76e6T^{2} \)
7 \( 1 + 2.26e4iT - 2.82e8T^{2} \)
11 \( 1 - 7.73e4iT - 2.59e10T^{2} \)
13 \( 1 + 3.19e5T + 1.37e11T^{2} \)
17 \( 1 - 2.42e6T + 2.01e12T^{2} \)
19 \( 1 + 1.56e6iT - 6.13e12T^{2} \)
23 \( 1 + 2.60e6iT - 4.14e13T^{2} \)
29 \( 1 + 9.86e6T + 4.20e14T^{2} \)
31 \( 1 + 4.14e7iT - 8.19e14T^{2} \)
37 \( 1 + 7.55e7T + 4.80e15T^{2} \)
41 \( 1 - 9.41e6T + 1.34e16T^{2} \)
43 \( 1 - 8.08e6iT - 2.16e16T^{2} \)
47 \( 1 - 1.70e8iT - 5.25e16T^{2} \)
53 \( 1 + 1.39e7T + 1.74e17T^{2} \)
59 \( 1 + 1.04e9iT - 5.11e17T^{2} \)
61 \( 1 + 6.24e8T + 7.13e17T^{2} \)
67 \( 1 + 1.71e9iT - 1.82e18T^{2} \)
71 \( 1 - 1.73e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.40e9T + 4.29e18T^{2} \)
79 \( 1 + 2.30e9iT - 9.46e18T^{2} \)
83 \( 1 - 2.97e7iT - 1.55e19T^{2} \)
89 \( 1 + 9.78e9T + 3.11e19T^{2} \)
97 \( 1 - 7.21e9T + 7.37e19T^{2} \)
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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

−16.83681609537000748023763632577, −15.37573215697213449694601525620, −14.28303939576109367298953099968, −12.44376866116833167487306426171, −11.20981206616835959313546740311, −9.940662493242188654024096604380, −7.49959383780954087308718111563, −4.63536039771453169674685413973, −3.52211358246456617170723073223, −0.41287979221803733959971612283, 3.27693799340222866447681489334, 5.45799866919511492209356164656, 7.41470323045901523784995030394, 8.515451207153957444372996124625, 11.91524215840464372819528835518, 12.34178108098924961236340055074, 14.42978735849049459289824505069, 15.46352353026659361851154343441, 16.61662674230135052579190100918, 18.44459819312422890138182587084

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