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Properties

Label 2-12-4.3-c20-0-14
Degree $2$
Conductor $12$
Sign $-0.336 + 0.941i$
Analytic cond. $30.4216$
Root an. cond. $5.51558$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.00e3 − 174. i)2-s + 3.40e4i·3-s + (9.87e5 + 3.52e5i)4-s + 5.90e6·5-s + (5.95e6 − 3.43e7i)6-s − 2.05e8i·7-s + (−9.34e8 − 5.28e8i)8-s − 1.16e9·9-s + (−5.95e9 − 1.03e9i)10-s − 6.76e9i·11-s + (−1.20e10 + 3.36e10i)12-s + 1.26e10·13-s + (−3.59e10 + 2.07e11i)14-s + 2.01e11i·15-s + (8.51e11 + 6.96e11i)16-s − 4.23e11·17-s + ⋯
L(s)  = 1  + (−0.985 − 0.170i)2-s + 0.577i·3-s + (0.941 + 0.336i)4-s + 0.604·5-s + (0.0984 − 0.568i)6-s − 0.729i·7-s + (−0.870 − 0.491i)8-s − 0.333·9-s + (−0.595 − 0.103i)10-s − 0.260i·11-s + (−0.194 + 0.543i)12-s + 0.0915·13-s + (−0.124 + 0.718i)14-s + 0.349i·15-s + (0.774 + 0.633i)16-s − 0.210·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.336 + 0.941i$
Analytic conductor: \(30.4216\)
Root analytic conductor: \(5.51558\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :10),\ -0.336 + 0.941i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.7106960066\)
\(L(\frac12)\) \(\approx\) \(0.7106960066\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.00e3 + 174. i)T \)
3 \( 1 - 3.40e4iT \)
good5 \( 1 - 5.90e6T + 9.53e13T^{2} \)
7 \( 1 + 2.05e8iT - 7.97e16T^{2} \)
11 \( 1 + 6.76e9iT - 6.72e20T^{2} \)
13 \( 1 - 1.26e10T + 1.90e22T^{2} \)
17 \( 1 + 4.23e11T + 4.06e24T^{2} \)
19 \( 1 - 2.41e12iT - 3.75e25T^{2} \)
23 \( 1 - 2.16e13iT - 1.71e27T^{2} \)
29 \( 1 + 5.86e14T + 1.76e29T^{2} \)
31 \( 1 + 1.62e15iT - 6.71e29T^{2} \)
37 \( 1 - 2.36e15T + 2.31e31T^{2} \)
41 \( 1 + 5.51e15T + 1.80e32T^{2} \)
43 \( 1 + 2.09e16iT - 4.67e32T^{2} \)
47 \( 1 + 4.99e16iT - 2.76e33T^{2} \)
53 \( 1 + 2.14e17T + 3.05e34T^{2} \)
59 \( 1 - 5.17e17iT - 2.61e35T^{2} \)
61 \( 1 - 1.08e18T + 5.08e35T^{2} \)
67 \( 1 + 2.98e18iT - 3.32e36T^{2} \)
71 \( 1 + 6.13e18iT - 1.05e37T^{2} \)
73 \( 1 + 4.46e18T + 1.84e37T^{2} \)
79 \( 1 + 1.16e19iT - 8.96e37T^{2} \)
83 \( 1 + 1.74e19iT - 2.40e38T^{2} \)
89 \( 1 + 3.64e19T + 9.72e38T^{2} \)
97 \( 1 - 2.48e19T + 5.43e39T^{2} \)
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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

−15.13329752115855503162933379031, −13.42595747448440092834284381388, −11.47511050819658165229299727226, −10.27399739466509503057476407269, −9.254457989394952786699271736962, −7.64449737826568749835885376429, −5.93444236614215954627878609900, −3.71312080699340676451397800583, −1.94736347532878490150446147997, −0.29879800369480930436161166290, 1.42439457908273137521441305227, 2.58438272881835512527103282871, 5.61715070729059615214875231244, 6.89029451097968365515180460634, 8.441634084421260969016840626437, 9.656492873144867678795141832971, 11.26248031698700126922273526757, 12.69577034244794399703448854721, 14.46621086604498267219055103526, 15.87433253267510937580245532453

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