↑

Properties

Label 2-12-4.3-c12-0-10
Degree $2$
Conductor $12$
Sign $-0.817 + 0.575i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (29.4 − 56.8i)2-s − 420. i·3-s + (−2.35e3 − 3.34e3i)4-s + 2.89e4·5-s + (−2.39e4 − 1.24e4i)6-s − 1.57e5i·7-s + (−2.59e5 + 3.51e4i)8-s − 1.77e5·9-s + (8.53e5 − 1.64e6i)10-s + 1.55e6i·11-s + (−1.40e6 + 9.92e5i)12-s − 4.64e6·13-s + (−8.97e6 − 4.65e6i)14-s − 1.21e7i·15-s + (−5.65e6 + 1.57e7i)16-s + 9.00e6·17-s + ⋯
L(s)  = 1  + (0.460 − 0.887i)2-s − 0.577i·3-s + (−0.575 − 0.817i)4-s + 1.85·5-s + (−0.512 − 0.265i)6-s − 1.34i·7-s + (−0.990 + 0.134i)8-s − 0.333·9-s + (0.853 − 1.64i)10-s + 0.879i·11-s + (−0.472 + 0.332i)12-s − 0.962·13-s + (−1.19 − 0.618i)14-s − 1.06i·15-s + (−0.337 + 0.941i)16-s + 0.372·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.817 + 0.575i$
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: \((2,\ 12,\ (\ :6),\ -0.817 + 0.575i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.766054 - 2.41910i\)
\(L(\frac12)\) \(\approx\) \(0.766054 - 2.41910i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-29.4 + 56.8i)T \)
3 \( 1 + 420. iT \)
good5 \( 1 - 2.89e4T + 2.44e8T^{2} \)
7 \( 1 + 1.57e5iT - 1.38e10T^{2} \)
11 \( 1 - 1.55e6iT - 3.13e12T^{2} \)
13 \( 1 + 4.64e6T + 2.32e13T^{2} \)
17 \( 1 - 9.00e6T + 5.82e14T^{2} \)
19 \( 1 + 4.57e7iT - 2.21e15T^{2} \)
23 \( 1 - 7.97e6iT - 2.19e16T^{2} \)
29 \( 1 - 8.09e8T + 3.53e17T^{2} \)
31 \( 1 - 4.31e8iT - 7.87e17T^{2} \)
37 \( 1 + 1.23e9T + 6.58e18T^{2} \)
41 \( 1 - 3.60e9T + 2.25e19T^{2} \)
43 \( 1 + 1.36e8iT - 3.99e19T^{2} \)
47 \( 1 - 5.87e9iT - 1.16e20T^{2} \)
53 \( 1 - 6.42e9T + 4.91e20T^{2} \)
59 \( 1 + 4.44e10iT - 1.77e21T^{2} \)
61 \( 1 - 1.83e10T + 2.65e21T^{2} \)
67 \( 1 - 6.73e10iT - 8.18e21T^{2} \)
71 \( 1 + 9.67e10iT - 1.64e22T^{2} \)
73 \( 1 - 1.46e11T + 2.29e22T^{2} \)
79 \( 1 + 9.30e10iT - 5.90e22T^{2} \)
83 \( 1 - 5.31e11iT - 1.06e23T^{2} \)
89 \( 1 + 1.05e11T + 2.46e23T^{2} \)
97 \( 1 - 7.32e11T + 6.93e23T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.28848113481449778721436439638, −14.38528662836449245808053488372, −13.62111477662230257702905921204, −12.56366369697338759836153907927, −10.51564475775248391710284525546, −9.571375440584077733694297480447, −6.74894212580766195607638580757, −4.95627984622138324219539254014, −2.43680522867133081611075224131, −1.08220836331437998488158736278, 2.65156154114481790958712462019, 5.28594498703015877355729623494, 6.08912504123053581651051084479, 8.704374635772965845468197892154, 9.859036885942647308383590660547, 12.33381070290364421653461683536, 13.85895898844825468970223539979, 14.81020819556826195469778693317, 16.34108874622826152937617978756, 17.41900621874252318932976245780

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