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Properties

Label 2-12-4.3-c18-0-14
Degree $2$
Conductor $12$
Sign $0.284 + 0.958i$
Analytic cond. $24.6463$
Root an. cond. $4.96450$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (506. − 73.5i)2-s − 1.13e4i·3-s + (2.51e5 − 7.44e4i)4-s + 2.11e6·5-s + (−8.35e5 − 5.75e6i)6-s − 4.00e7i·7-s + (1.21e8 − 5.62e7i)8-s − 1.29e8·9-s + (1.07e9 − 1.55e8i)10-s + 2.33e9i·11-s + (−8.46e8 − 2.85e9i)12-s + 4.89e9·13-s + (−2.94e9 − 2.03e10i)14-s − 2.40e10i·15-s + (5.76e10 − 3.74e10i)16-s − 2.82e10·17-s + ⋯
L(s)  = 1  + (0.989 − 0.143i)2-s − 0.577i·3-s + (0.958 − 0.284i)4-s + 1.08·5-s + (−0.0828 − 0.571i)6-s − 0.993i·7-s + (0.908 − 0.418i)8-s − 0.333·9-s + (1.07 − 0.155i)10-s + 0.988i·11-s + (−0.164 − 0.553i)12-s + 0.461·13-s + (−0.142 − 0.983i)14-s − 0.625i·15-s + (0.838 − 0.544i)16-s − 0.238·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.284 + 0.958i$
Analytic conductor: \(24.6463\)
Root analytic conductor: \(4.96450\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :9),\ 0.284 + 0.958i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(4.490093109\)
\(L(\frac12)\) \(\approx\) \(4.490093109\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-506. + 73.5i)T \)
3 \( 1 + 1.13e4iT \)
good5 \( 1 - 2.11e6T + 3.81e12T^{2} \)
7 \( 1 + 4.00e7iT - 1.62e15T^{2} \)
11 \( 1 - 2.33e9iT - 5.55e18T^{2} \)
13 \( 1 - 4.89e9T + 1.12e20T^{2} \)
17 \( 1 + 2.82e10T + 1.40e22T^{2} \)
19 \( 1 + 4.35e11iT - 1.04e23T^{2} \)
23 \( 1 - 7.05e10iT - 3.24e24T^{2} \)
29 \( 1 + 2.17e12T + 2.10e26T^{2} \)
31 \( 1 + 3.60e13iT - 6.99e26T^{2} \)
37 \( 1 - 2.32e14T + 1.68e28T^{2} \)
41 \( 1 + 6.17e14T + 1.07e29T^{2} \)
43 \( 1 - 1.00e15iT - 2.52e29T^{2} \)
47 \( 1 - 1.05e15iT - 1.25e30T^{2} \)
53 \( 1 - 1.66e15T + 1.08e31T^{2} \)
59 \( 1 - 1.58e16iT - 7.50e31T^{2} \)
61 \( 1 - 7.61e13T + 1.36e32T^{2} \)
67 \( 1 - 1.27e16iT - 7.40e32T^{2} \)
71 \( 1 + 8.50e15iT - 2.10e33T^{2} \)
73 \( 1 + 8.52e16T + 3.46e33T^{2} \)
79 \( 1 - 2.56e16iT - 1.43e34T^{2} \)
83 \( 1 - 3.31e17iT - 3.49e34T^{2} \)
89 \( 1 - 1.22e17T + 1.22e35T^{2} \)
97 \( 1 + 7.04e15T + 5.77e35T^{2} \)
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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

−14.99366213665569375042573519544, −13.60193354731622175961205863607, −13.07475253482091904696105733588, −11.30329349283351613865693722161, −9.847762259325751047649695100906, −7.31589801759544516264407313526, −6.13142397378034563604016868900, −4.47137752928442734195705099983, −2.49144373264388641152723130754, −1.19694692268402293551304930518, 1.94781812339676251842592685015, 3.42205018056497293045714718992, 5.35018318941620241739302913215, 6.16562020342044150246342194009, 8.608359497036028039095955220648, 10.35330228273546575512410951685, 11.85667008964347120079296370944, 13.39713996238012246772731518490, 14.47597012259489618420709622038, 15.78161381853682136573410959787

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