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Properties

Label 2-12-4.3-c12-0-7
Degree $2$
Conductor $12$
Sign $0.920 - 0.391i$
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  + (−35.2 + 53.3i)2-s + 420. i·3-s + (−1.60e3 − 3.76e3i)4-s + 8.40e3·5-s + (−2.24e4 − 1.48e4i)6-s − 1.92e5i·7-s + (2.57e5 + 4.73e4i)8-s − 1.77e5·9-s + (−2.96e5 + 4.48e5i)10-s + 7.48e5i·11-s + (1.58e6 − 6.75e5i)12-s + 5.80e6·13-s + (1.02e7 + 6.77e6i)14-s + 3.53e6i·15-s + (−1.16e7 + 1.20e7i)16-s + 2.57e7·17-s + ⋯
L(s)  = 1  + (−0.551 + 0.834i)2-s + 0.577i·3-s + (−0.391 − 0.920i)4-s + 0.538·5-s + (−0.481 − 0.318i)6-s − 1.63i·7-s + (0.983 + 0.180i)8-s − 0.333·9-s + (−0.296 + 0.448i)10-s + 0.422i·11-s + (0.531 − 0.226i)12-s + 1.20·13-s + (1.36 + 0.900i)14-s + 0.310i·15-s + (−0.692 + 0.721i)16-s + 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.920 - 0.391i$
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: \((2,\ 12,\ (\ :6),\ 0.920 - 0.391i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.38639 + 0.282932i\)
\(L(\frac12)\) \(\approx\) \(1.38639 + 0.282932i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (35.2 - 53.3i)T \)
3 \( 1 - 420. iT \)
good5 \( 1 - 8.40e3T + 2.44e8T^{2} \)
7 \( 1 + 1.92e5iT - 1.38e10T^{2} \)
11 \( 1 - 7.48e5iT - 3.13e12T^{2} \)
13 \( 1 - 5.80e6T + 2.32e13T^{2} \)
17 \( 1 - 2.57e7T + 5.82e14T^{2} \)
19 \( 1 - 1.62e7iT - 2.21e15T^{2} \)
23 \( 1 + 1.99e8iT - 2.19e16T^{2} \)
29 \( 1 - 8.06e8T + 3.53e17T^{2} \)
31 \( 1 + 7.40e8iT - 7.87e17T^{2} \)
37 \( 1 - 4.40e9T + 6.58e18T^{2} \)
41 \( 1 - 3.32e9T + 2.25e19T^{2} \)
43 \( 1 - 4.43e9iT - 3.99e19T^{2} \)
47 \( 1 + 1.18e10iT - 1.16e20T^{2} \)
53 \( 1 + 1.52e10T + 4.91e20T^{2} \)
59 \( 1 - 2.87e10iT - 1.77e21T^{2} \)
61 \( 1 + 5.04e10T + 2.65e21T^{2} \)
67 \( 1 + 7.04e10iT - 8.18e21T^{2} \)
71 \( 1 - 4.83e10iT - 1.64e22T^{2} \)
73 \( 1 - 2.79e10T + 2.29e22T^{2} \)
79 \( 1 + 4.03e11iT - 5.90e22T^{2} \)
83 \( 1 - 3.84e11iT - 1.06e23T^{2} \)
89 \( 1 + 4.78e11T + 2.46e23T^{2} \)
97 \( 1 + 1.21e12T + 6.93e23T^{2} \)
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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

−17.00628852439263580793419205592, −16.21532219477268396287023773110, −14.54555880819426636859535288445, −13.53222035809953924423832131596, −10.68663355004914528547473543385, −9.777055416672635562081066960964, −7.940982587133454379315821909273, −6.23074103755793946645854159927, −4.28615694760774857728803385371, −0.942137533631442017383911886882, 1.37633131174062125537607213690, 2.91087830751793004827704968477, 5.83610488727463270232639836707, 8.218868625462873562131629163852, 9.432788177604755101846413887730, 11.34644567846775351164946056542, 12.49201932860072540883720881570, 13.81515123563165873886502661953, 15.92139181536608776483797669550, 17.66985953178370781250064106011

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