Properties

Label 2-12-4.3-c2-0-0
Degree $2$
Conductor $12$
Sign $0.866 - 0.5i$
Analytic cond. $0.326976$
Root an. cond. $0.571818$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s − 1.73i·3-s + (−1.99 − 3.46i)4-s − 2·5-s + (2.99 + 1.73i)6-s + 6.92i·7-s + 7.99·8-s − 2.99·9-s + (2 − 3.46i)10-s − 6.92i·11-s + (−5.99 + 3.46i)12-s + 2·13-s + (−11.9 − 6.92i)14-s + 3.46i·15-s + (−8 + 13.8i)16-s + 10·17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s − 0.577i·3-s + (−0.499 − 0.866i)4-s − 0.400·5-s + (0.499 + 0.288i)6-s + 0.989i·7-s + 0.999·8-s − 0.333·9-s + (0.200 − 0.346i)10-s − 0.629i·11-s + (−0.499 + 0.288i)12-s + 0.153·13-s + (−0.857 − 0.494i)14-s + 0.230i·15-s + (−0.5 + 0.866i)16-s + 0.588·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(0.326976\)
Root analytic conductor: \(0.571818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :1),\ 0.866 - 0.5i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.541943 + 0.145213i\)
\(L(\frac12)\) \(\approx\) \(0.541943 + 0.145213i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 + 1.73iT \)
good5 \( 1 + 2T + 25T^{2} \)
7 \( 1 - 6.92iT - 49T^{2} \)
11 \( 1 + 6.92iT - 121T^{2} \)
13 \( 1 - 2T + 169T^{2} \)
17 \( 1 - 10T + 289T^{2} \)
19 \( 1 + 20.7iT - 361T^{2} \)
23 \( 1 - 27.7iT - 529T^{2} \)
29 \( 1 + 26T + 841T^{2} \)
31 \( 1 + 6.92iT - 961T^{2} \)
37 \( 1 - 26T + 1.36e3T^{2} \)
41 \( 1 - 58T + 1.68e3T^{2} \)
43 \( 1 - 48.4iT - 1.84e3T^{2} \)
47 \( 1 + 69.2iT - 2.20e3T^{2} \)
53 \( 1 + 74T + 2.80e3T^{2} \)
59 \( 1 - 90.0iT - 3.48e3T^{2} \)
61 \( 1 - 26T + 3.72e3T^{2} \)
67 \( 1 + 6.92iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 46T + 5.32e3T^{2} \)
79 \( 1 + 117. iT - 6.24e3T^{2} \)
83 \( 1 + 48.4iT - 6.88e3T^{2} \)
89 \( 1 - 82T + 7.92e3T^{2} \)
97 \( 1 - 2T + 9.40e3T^{2} \)
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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

−19.63454363885461930623749185310, −18.74904406917276109914630132479, −17.62690934090527446578915444830, −16.10483807261337693382380385118, −14.93792911843039893260472154080, −13.35787087310893432453946414938, −11.45876558534976765996277661143, −9.174383214369402603361117159153, −7.71002150047828499977504048565, −5.81330292801615233284569326855, 4.04688556322157586076116458372, 7.82005355956976890465359257184, 9.751538032754624187969349089619, 10.93409908939634533731498037515, 12.51798678506732196418498026374, 14.27703873788863352896282392074, 16.24609613826692652410721547212, 17.33981884772548023069555570954, 18.88346699411507880536372215953, 20.21368534244247546625260313645

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