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Properties

Label 2-2e4-16.3-c12-0-11
Degree $2$
Conductor $16$
Sign $-0.0603 - 0.998i$
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  + (−48.8 + 41.3i)2-s + (−913. + 913. i)3-s + (677. − 4.03e3i)4-s + (70.8 − 70.8i)5-s + (6.86e3 − 8.23e4i)6-s + 2.25e5·7-s + (1.33e5 + 2.25e5i)8-s − 1.13e6i·9-s + (−531. + 6.38e3i)10-s + (8.24e3 + 8.24e3i)11-s + (3.07e6 + 4.30e6i)12-s + (−2.18e6 − 2.18e6i)13-s + (−1.10e7 + 9.34e6i)14-s + 1.29e5i·15-s + (−1.58e7 − 5.47e6i)16-s + 3.07e7·17-s + ⋯
L(s)  = 1  + (−0.763 + 0.645i)2-s + (−1.25 + 1.25i)3-s + (0.165 − 0.986i)4-s + (0.00453 − 0.00453i)5-s + (0.147 − 1.76i)6-s + 1.92·7-s + (0.510 + 0.859i)8-s − 2.13i·9-s + (−0.000531 + 0.00638i)10-s + (0.00465 + 0.00465i)11-s + (1.02 + 1.44i)12-s + (−0.453 − 0.453i)13-s + (−1.46 + 1.24i)14-s + 0.0113i·15-s + (−0.945 − 0.326i)16-s + 1.27·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0603 - 0.998i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.0603 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.0603 - 0.998i$
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: \((2,\ 16,\ (\ :6),\ -0.0603 - 0.998i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.678728 + 0.720972i\)
\(L(\frac12)\) \(\approx\) \(0.678728 + 0.720972i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (48.8 - 41.3i)T \)
good3 \( 1 + (913. - 913. i)T - 5.31e5iT^{2} \)
5 \( 1 + (-70.8 + 70.8i)T - 2.44e8iT^{2} \)
7 \( 1 - 2.25e5T + 1.38e10T^{2} \)
11 \( 1 + (-8.24e3 - 8.24e3i)T + 3.13e12iT^{2} \)
13 \( 1 + (2.18e6 + 2.18e6i)T + 2.32e13iT^{2} \)
17 \( 1 - 3.07e7T + 5.82e14T^{2} \)
19 \( 1 + (-1.32e7 + 1.32e7i)T - 2.21e15iT^{2} \)
23 \( 1 - 1.29e8T + 2.19e16T^{2} \)
29 \( 1 + (5.71e7 + 5.71e7i)T + 3.53e17iT^{2} \)
31 \( 1 + 1.28e9iT - 7.87e17T^{2} \)
37 \( 1 + (-3.59e8 + 3.59e8i)T - 6.58e18iT^{2} \)
41 \( 1 - 5.63e9iT - 2.25e19T^{2} \)
43 \( 1 + (1.03e7 + 1.03e7i)T + 3.99e19iT^{2} \)
47 \( 1 - 5.79e9iT - 1.16e20T^{2} \)
53 \( 1 + (1.73e10 - 1.73e10i)T - 4.91e20iT^{2} \)
59 \( 1 + (-3.92e10 - 3.92e10i)T + 1.77e21iT^{2} \)
61 \( 1 + (4.35e10 + 4.35e10i)T + 2.65e21iT^{2} \)
67 \( 1 + (-5.24e10 + 5.24e10i)T - 8.18e21iT^{2} \)
71 \( 1 - 9.46e10T + 1.64e22T^{2} \)
73 \( 1 - 1.03e11iT - 2.29e22T^{2} \)
79 \( 1 - 1.29e11iT - 5.90e22T^{2} \)
83 \( 1 + (-2.86e11 + 2.86e11i)T - 1.06e23iT^{2} \)
89 \( 1 - 2.87e10iT - 2.46e23T^{2} \)
97 \( 1 + 1.90e11T + 6.93e23T^{2} \)
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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

−16.84502620517345537432530043020, −15.33003974312439791853059002319, −14.61708814178939396135130625864, −11.58175396456968593346832161230, −10.82119788485915564459219154737, −9.519733016328814563034490766231, −7.74708652192758918368211154166, −5.59411355851099103448165886544, −4.76947844989598944542152150452, −0.986020551456430594945505200934, 0.899758055532210656026104979332, 1.86839139605124747159294388069, 5.07831886093477878305955798665, 7.15657042145224968443446628612, 8.203209430970071573440784784732, 10.62873242621033666884349840825, 11.66673387441201865574356425017, 12.36733321435092338472642682203, 14.12235718021482675671295067321, 16.61658606005598404794819885319

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