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Properties

Label 2-12-4.3-c12-0-1
Degree $2$
Conductor $12$
Sign $0.434 - 0.900i$
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  + (−62.3 − 14.2i)2-s − 420. i·3-s + (3.68e3 + 1.78e3i)4-s − 2.16e4·5-s + (−6.00e3 + 2.62e4i)6-s − 1.55e5i·7-s + (−2.04e5 − 1.63e5i)8-s − 1.77e5·9-s + (1.34e6 + 3.08e5i)10-s + 3.12e6i·11-s + (7.49e5 − 1.55e6i)12-s + 1.10e6·13-s + (−2.21e6 + 9.68e6i)14-s + 9.10e6i·15-s + (1.04e7 + 1.31e7i)16-s + 3.10e7·17-s + ⋯
L(s)  = 1  + (−0.974 − 0.223i)2-s − 0.577i·3-s + (0.900 + 0.434i)4-s − 1.38·5-s + (−0.128 + 0.562i)6-s − 1.31i·7-s + (−0.780 − 0.624i)8-s − 0.333·9-s + (1.34 + 0.308i)10-s + 1.76i·11-s + (0.251 − 0.519i)12-s + 0.228·13-s + (−0.294 + 1.28i)14-s + 0.798i·15-s + (0.621 + 0.783i)16-s + 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.434 - 0.900i$
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: \((2,\ 12,\ (\ :6),\ 0.434 - 0.900i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.380601 + 0.238876i\)
\(L(\frac12)\) \(\approx\) \(0.380601 + 0.238876i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (62.3 + 14.2i)T \)
3 \( 1 + 420. iT \)
good5 \( 1 + 2.16e4T + 2.44e8T^{2} \)
7 \( 1 + 1.55e5iT - 1.38e10T^{2} \)
11 \( 1 - 3.12e6iT - 3.13e12T^{2} \)
13 \( 1 - 1.10e6T + 2.32e13T^{2} \)
17 \( 1 - 3.10e7T + 5.82e14T^{2} \)
19 \( 1 - 2.82e7iT - 2.21e15T^{2} \)
23 \( 1 - 1.30e8iT - 2.19e16T^{2} \)
29 \( 1 + 7.45e8T + 3.53e17T^{2} \)
31 \( 1 - 3.42e8iT - 7.87e17T^{2} \)
37 \( 1 - 1.27e9T + 6.58e18T^{2} \)
41 \( 1 + 8.57e8T + 2.25e19T^{2} \)
43 \( 1 - 6.63e9iT - 3.99e19T^{2} \)
47 \( 1 + 7.75e9iT - 1.16e20T^{2} \)
53 \( 1 + 1.80e10T + 4.91e20T^{2} \)
59 \( 1 + 6.86e9iT - 1.77e21T^{2} \)
61 \( 1 - 2.30e10T + 2.65e21T^{2} \)
67 \( 1 - 1.09e11iT - 8.18e21T^{2} \)
71 \( 1 - 2.98e10iT - 1.64e22T^{2} \)
73 \( 1 + 2.57e11T + 2.29e22T^{2} \)
79 \( 1 - 3.42e11iT - 5.90e22T^{2} \)
83 \( 1 + 5.68e10iT - 1.06e23T^{2} \)
89 \( 1 - 6.63e11T + 2.46e23T^{2} \)
97 \( 1 + 9.94e11T + 6.93e23T^{2} \)
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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

−17.51803908753072953952602099408, −16.36290849317211234855155665206, −14.88118756142163747599890839448, −12.64985482602895985804865068815, −11.50336258293104047644766423690, −9.983231475262399082878839932350, −7.78468345145206739555293990503, −7.24390834351371214833901658936, −3.75765091641612526520270160705, −1.28563939032096444202828659286, 0.31369821404862009919089008355, 3.15573045392254991552928718793, 5.77736618512218812365725679403, 8.027382968078493043183658916284, 9.004460555629182822207724539342, 10.99904865884171900354238807304, 11.92560223632206823300099073158, 14.78941583657195403525419376066, 15.81009171349001517876688051231, 16.58335511848992166062182188313

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