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Properties

Label 2-12-4.3-c12-0-11
Degree $2$
Conductor $12$
Sign $-0.958 - 0.286i$
Analytic cond. $10.9679$
Root an. cond. $3.31178$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (51.3 − 38.2i)2-s − 420. i·3-s + (1.17e3 − 3.92e3i)4-s − 2.45e4·5-s + (−1.60e4 − 2.16e4i)6-s + 9.64e4i·7-s + (−8.99e4 − 2.46e5i)8-s − 1.77e5·9-s + (−1.26e6 + 9.39e5i)10-s − 5.82e5i·11-s + (−1.65e6 − 4.93e5i)12-s + 8.90e5·13-s + (3.68e6 + 4.95e6i)14-s + 1.03e7i·15-s + (−1.40e7 − 9.19e6i)16-s − 4.03e7·17-s + ⋯
L(s)  = 1  + (0.801 − 0.597i)2-s − 0.577i·3-s + (0.286 − 0.958i)4-s − 1.57·5-s + (−0.344 − 0.462i)6-s + 0.820i·7-s + (−0.343 − 0.939i)8-s − 0.333·9-s + (−1.26 + 0.939i)10-s − 0.328i·11-s + (−0.553 − 0.165i)12-s + 0.184·13-s + (0.489 + 0.657i)14-s + 0.907i·15-s + (−0.836 − 0.548i)16-s − 1.67·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.958 - 0.286i$
Analytic conductor: \(10.9679\)
Root analytic conductor: \(3.31178\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :6),\ -0.958 - 0.286i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.155202 + 1.06236i\)
\(L(\frac12)\) \(\approx\) \(0.155202 + 1.06236i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-51.3 + 38.2i)T \)
3 \( 1 + 420. iT \)
good5 \( 1 + 2.45e4T + 2.44e8T^{2} \)
7 \( 1 - 9.64e4iT - 1.38e10T^{2} \)
11 \( 1 + 5.82e5iT - 3.13e12T^{2} \)
13 \( 1 - 8.90e5T + 2.32e13T^{2} \)
17 \( 1 + 4.03e7T + 5.82e14T^{2} \)
19 \( 1 + 4.37e7iT - 2.21e15T^{2} \)
23 \( 1 + 1.62e8iT - 2.19e16T^{2} \)
29 \( 1 - 9.58e8T + 3.53e17T^{2} \)
31 \( 1 + 1.21e9iT - 7.87e17T^{2} \)
37 \( 1 - 8.12e7T + 6.58e18T^{2} \)
41 \( 1 + 4.04e9T + 2.25e19T^{2} \)
43 \( 1 - 1.25e10iT - 3.99e19T^{2} \)
47 \( 1 + 1.73e10iT - 1.16e20T^{2} \)
53 \( 1 + 1.80e10T + 4.91e20T^{2} \)
59 \( 1 - 3.40e10iT - 1.77e21T^{2} \)
61 \( 1 - 2.69e10T + 2.65e21T^{2} \)
67 \( 1 - 1.34e10iT - 8.18e21T^{2} \)
71 \( 1 - 2.95e9iT - 1.64e22T^{2} \)
73 \( 1 - 1.41e10T + 2.29e22T^{2} \)
79 \( 1 - 9.53e10iT - 5.90e22T^{2} \)
83 \( 1 + 4.73e11iT - 1.06e23T^{2} \)
89 \( 1 + 1.72e11T + 2.46e23T^{2} \)
97 \( 1 + 3.98e11T + 6.93e23T^{2} \)
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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

−15.87895755279967869189107590923, −14.99584006089916757473748985477, −13.24087128561046059315093549196, −11.98200247693981214237341521638, −11.14743584535562582702618383522, −8.576790103599798523731444592691, −6.60697635094981621274472603549, −4.47949424677725997956884646613, −2.66772379260223240715877082697, −0.36708071654063517238260569160, 3.58186612343628607076059231808, 4.59495438121885909834065912026, 6.98686290125048788340801246190, 8.359684770534571853252963433285, 10.93507997408946805634782978740, 12.20721003473700969628487833452, 13.92852445088517491874360137221, 15.40719003698576664079290268971, 16.00904312169558572052435341730, 17.44840127813487438450488209243

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