Properties

Label 2-2e4-16.11-c8-0-0
Degree $2$
Conductor $16$
Sign $-0.965 - 0.260i$
Analytic cond. $6.51805$
Root an. cond. $2.55304$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−14.5 + 6.59i)2-s + (−7.14 − 7.14i)3-s + (168. − 192. i)4-s + (610. + 610. i)5-s + (151. + 56.9i)6-s − 3.41e3·7-s + (−1.19e3 + 3.91e3i)8-s − 6.45e3i·9-s + (−1.29e4 − 4.87e3i)10-s + (−1.17e4 + 1.17e4i)11-s + (−2.57e3 + 166. i)12-s + (−3.58e4 + 3.58e4i)13-s + (4.97e4 − 2.24e4i)14-s − 8.72e3i·15-s + (−8.44e3 − 6.49e4i)16-s + 2.60e4·17-s + ⋯
L(s)  = 1  + (−0.911 + 0.412i)2-s + (−0.0881 − 0.0881i)3-s + (0.659 − 0.751i)4-s + (0.977 + 0.977i)5-s + (0.116 + 0.0439i)6-s − 1.42·7-s + (−0.291 + 0.956i)8-s − 0.984i·9-s + (−1.29 − 0.487i)10-s + (−0.804 + 0.804i)11-s + (−0.124 + 0.00805i)12-s + (−1.25 + 1.25i)13-s + (1.29 − 0.585i)14-s − 0.172i·15-s + (−0.128 − 0.991i)16-s + 0.312·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.965 - 0.260i$
Analytic conductor: \(6.51805\)
Root analytic conductor: \(2.55304\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :4),\ -0.965 - 0.260i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0590214 + 0.445541i\)
\(L(\frac12)\) \(\approx\) \(0.0590214 + 0.445541i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (14.5 - 6.59i)T \)
good3 \( 1 + (7.14 + 7.14i)T + 6.56e3iT^{2} \)
5 \( 1 + (-610. - 610. i)T + 3.90e5iT^{2} \)
7 \( 1 + 3.41e3T + 5.76e6T^{2} \)
11 \( 1 + (1.17e4 - 1.17e4i)T - 2.14e8iT^{2} \)
13 \( 1 + (3.58e4 - 3.58e4i)T - 8.15e8iT^{2} \)
17 \( 1 - 2.60e4T + 6.97e9T^{2} \)
19 \( 1 + (-2.44e4 - 2.44e4i)T + 1.69e10iT^{2} \)
23 \( 1 + 3.94e5T + 7.83e10T^{2} \)
29 \( 1 + (-2.48e5 + 2.48e5i)T - 5.00e11iT^{2} \)
31 \( 1 - 3.19e5iT - 8.52e11T^{2} \)
37 \( 1 + (-1.76e6 - 1.76e6i)T + 3.51e12iT^{2} \)
41 \( 1 - 7.60e4iT - 7.98e12T^{2} \)
43 \( 1 + (-1.18e6 + 1.18e6i)T - 1.16e13iT^{2} \)
47 \( 1 + 2.40e6iT - 2.38e13T^{2} \)
53 \( 1 + (-5.51e6 - 5.51e6i)T + 6.22e13iT^{2} \)
59 \( 1 + (-3.44e6 + 3.44e6i)T - 1.46e14iT^{2} \)
61 \( 1 + (6.76e6 - 6.76e6i)T - 1.91e14iT^{2} \)
67 \( 1 + (1.44e7 + 1.44e7i)T + 4.06e14iT^{2} \)
71 \( 1 - 3.76e7T + 6.45e14T^{2} \)
73 \( 1 - 9.13e6iT - 8.06e14T^{2} \)
79 \( 1 - 6.48e7iT - 1.51e15T^{2} \)
83 \( 1 + (9.64e6 + 9.64e6i)T + 2.25e15iT^{2} \)
89 \( 1 + 5.71e7iT - 3.93e15T^{2} \)
97 \( 1 + 8.72e7T + 7.83e15T^{2} \)
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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

−17.92612383566043695783509839900, −16.71340718560275003054819817321, −15.31146385581384615263404834898, −14.09822074010616484983425168642, −12.16475736720374136280298177316, −10.01050323380159464785227618380, −9.651862353621217941412082812187, −7.07933900734774353287985103741, −6.16648449364496450884688353639, −2.44561741765901249633260810235, 0.30953155581315455588021513289, 2.61366588289617394419872726027, 5.67586062563530578396731015043, 7.922479586443778251390013356040, 9.567616898746727532291790406927, 10.39406467509461519679499079668, 12.54914731387536153564094011244, 13.34193184699727712417335216541, 16.02013580532105914595972992000, 16.63274491739355244191476501064

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