Properties

Label 2-2e4-16.3-c12-0-17
Degree $2$
Conductor $16$
Sign $0.567 + 0.823i$
Analytic cond. $14.6239$
Root an. cond. $3.82412$
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  + (3.36 + 63.9i)2-s + (837. − 837. i)3-s + (−4.07e3 + 429. i)4-s + (5.67e3 − 5.67e3i)5-s + (5.63e4 + 5.06e4i)6-s − 4.58e3·7-s + (−4.11e4 − 2.58e5i)8-s − 8.70e5i·9-s + (3.81e5 + 3.43e5i)10-s + (−1.20e6 − 1.20e6i)11-s + (−3.05e6 + 3.77e6i)12-s + (−3.66e6 − 3.66e6i)13-s + (−1.54e4 − 2.93e5i)14-s − 9.49e6i·15-s + (1.64e7 − 3.50e6i)16-s + 2.38e7·17-s + ⋯
L(s)  = 1  + (0.0525 + 0.998i)2-s + (1.14 − 1.14i)3-s + (−0.994 + 0.104i)4-s + (0.362 − 0.362i)5-s + (1.20 + 1.08i)6-s − 0.0389·7-s + (−0.157 − 0.987i)8-s − 1.63i·9-s + (0.381 + 0.343i)10-s + (−0.681 − 0.681i)11-s + (−1.02 + 1.26i)12-s + (−0.759 − 0.759i)13-s + (−0.00204 − 0.0389i)14-s − 0.833i·15-s + (0.977 − 0.208i)16-s + 0.989·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.567 + 0.823i$
Analytic conductor: \(14.6239\)
Root analytic conductor: \(3.82412\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :6),\ 0.567 + 0.823i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.01222 - 1.05766i\)
\(L(\frac12)\) \(\approx\) \(2.01222 - 1.05766i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.36 - 63.9i)T \)
good3 \( 1 + (-837. + 837. i)T - 5.31e5iT^{2} \)
5 \( 1 + (-5.67e3 + 5.67e3i)T - 2.44e8iT^{2} \)
7 \( 1 + 4.58e3T + 1.38e10T^{2} \)
11 \( 1 + (1.20e6 + 1.20e6i)T + 3.13e12iT^{2} \)
13 \( 1 + (3.66e6 + 3.66e6i)T + 2.32e13iT^{2} \)
17 \( 1 - 2.38e7T + 5.82e14T^{2} \)
19 \( 1 + (-4.96e7 + 4.96e7i)T - 2.21e15iT^{2} \)
23 \( 1 + 1.13e8T + 2.19e16T^{2} \)
29 \( 1 + (-5.30e8 - 5.30e8i)T + 3.53e17iT^{2} \)
31 \( 1 + 1.47e9iT - 7.87e17T^{2} \)
37 \( 1 + (1.24e9 - 1.24e9i)T - 6.58e18iT^{2} \)
41 \( 1 + 7.97e9iT - 2.25e19T^{2} \)
43 \( 1 + (-3.00e9 - 3.00e9i)T + 3.99e19iT^{2} \)
47 \( 1 - 1.72e10iT - 1.16e20T^{2} \)
53 \( 1 + (6.60e9 - 6.60e9i)T - 4.91e20iT^{2} \)
59 \( 1 + (-2.56e10 - 2.56e10i)T + 1.77e21iT^{2} \)
61 \( 1 + (-3.97e10 - 3.97e10i)T + 2.65e21iT^{2} \)
67 \( 1 + (-5.14e10 + 5.14e10i)T - 8.18e21iT^{2} \)
71 \( 1 - 7.10e10T + 1.64e22T^{2} \)
73 \( 1 - 1.51e11iT - 2.29e22T^{2} \)
79 \( 1 - 2.84e11iT - 5.90e22T^{2} \)
83 \( 1 + (-1.63e10 + 1.63e10i)T - 1.06e23iT^{2} \)
89 \( 1 + 4.75e11iT - 2.46e23T^{2} \)
97 \( 1 + 7.25e11T + 6.93e23T^{2} \)
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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

−15.77524933673738729650585229771, −14.38082135252480570484752096682, −13.49944520436355131510127488195, −12.57121758402452832477546470654, −9.561657396528787289041901063614, −8.212576437735470407874916880441, −7.31020945154223035963913022894, −5.51026206210662001979535176073, −2.98137769574069149048037341250, −0.826534534763405392952726187815, 2.16456664001942397634306708717, 3.43198288593996876095863491466, 4.90227162946286113665542726545, 8.137309367993108381052112465057, 9.805573645086503544880618433667, 10.12801282139365495169990631184, 12.14723333044284284536971141529, 13.98496719224399197703446980563, 14.55083055599028582389734357043, 16.15362920609745782694321437290

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