Properties

Label 2-2e4-16.5-c7-0-8
Degree $2$
Conductor $16$
Sign $-0.333 + 0.942i$
Analytic cond. $4.99816$
Root an. cond. $2.23565$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.5 − 4.19i)2-s + (42.2 − 42.2i)3-s + (92.8 + 88.0i)4-s + (46.9 + 46.9i)5-s + (−620. + 266. i)6-s − 1.38e3i·7-s + (−606. − 1.31e3i)8-s − 1.37e3i·9-s + (−296. − 689. i)10-s + (−3.27e3 − 3.27e3i)11-s + (7.63e3 − 202. i)12-s + (−2.19e3 + 2.19e3i)13-s + (−5.80e3 + 1.45e4i)14-s + 3.96e3·15-s + (867. + 1.63e4i)16-s + 1.32e4·17-s + ⋯
L(s)  = 1  + (−0.928 − 0.370i)2-s + (0.902 − 0.902i)3-s + (0.725 + 0.688i)4-s + (0.167 + 0.167i)5-s + (−1.17 + 0.504i)6-s − 1.52i·7-s + (−0.419 − 0.907i)8-s − 0.629i·9-s + (−0.0937 − 0.218i)10-s + (−0.742 − 0.742i)11-s + (1.27 − 0.0337i)12-s + (−0.277 + 0.277i)13-s + (−0.565 + 1.41i)14-s + 0.303·15-s + (0.0529 + 0.998i)16-s + 0.653·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.333 + 0.942i$
Analytic conductor: \(4.99816\)
Root analytic conductor: \(2.23565\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :7/2),\ -0.333 + 0.942i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.740976 - 1.04779i\)
\(L(\frac12)\) \(\approx\) \(0.740976 - 1.04779i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (10.5 + 4.19i)T \)
good3 \( 1 + (-42.2 + 42.2i)T - 2.18e3iT^{2} \)
5 \( 1 + (-46.9 - 46.9i)T + 7.81e4iT^{2} \)
7 \( 1 + 1.38e3iT - 8.23e5T^{2} \)
11 \( 1 + (3.27e3 + 3.27e3i)T + 1.94e7iT^{2} \)
13 \( 1 + (2.19e3 - 2.19e3i)T - 6.27e7iT^{2} \)
17 \( 1 - 1.32e4T + 4.10e8T^{2} \)
19 \( 1 + (-3.45e4 + 3.45e4i)T - 8.93e8iT^{2} \)
23 \( 1 - 9.62e4iT - 3.40e9T^{2} \)
29 \( 1 + (8.06e4 - 8.06e4i)T - 1.72e10iT^{2} \)
31 \( 1 - 1.21e5T + 2.75e10T^{2} \)
37 \( 1 + (-2.09e5 - 2.09e5i)T + 9.49e10iT^{2} \)
41 \( 1 - 6.50e4iT - 1.94e11T^{2} \)
43 \( 1 + (-4.67e5 - 4.67e5i)T + 2.71e11iT^{2} \)
47 \( 1 + 4.98e5T + 5.06e11T^{2} \)
53 \( 1 + (-4.53e5 - 4.53e5i)T + 1.17e12iT^{2} \)
59 \( 1 + (7.95e5 + 7.95e5i)T + 2.48e12iT^{2} \)
61 \( 1 + (-1.35e6 + 1.35e6i)T - 3.14e12iT^{2} \)
67 \( 1 + (-1.63e6 + 1.63e6i)T - 6.06e12iT^{2} \)
71 \( 1 - 3.40e6iT - 9.09e12T^{2} \)
73 \( 1 - 3.34e6iT - 1.10e13T^{2} \)
79 \( 1 + 6.61e6T + 1.92e13T^{2} \)
83 \( 1 + (-3.95e6 + 3.95e6i)T - 2.71e13iT^{2} \)
89 \( 1 + 4.43e6iT - 4.42e13T^{2} \)
97 \( 1 + 1.21e7T + 8.07e13T^{2} \)
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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

−17.43120649900322618440681558917, −16.08992170317607762702509492463, −14.00079036942638192916197280458, −13.14076258579163289356877576502, −11.22697390278701098822415366873, −9.756764786911356440769167484614, −7.983051296196218727081299185182, −7.14123049466260510744818555532, −3.04195372246597100610508079132, −1.04976409624131233820455484175, 2.53369373453755996378962380468, 5.49676179465128058771437418872, 7.992208659434576660996039509660, 9.218957174111793780183118625344, 10.12500485905024325406035232294, 12.19150471310725242953684991638, 14.59118441673940946890427207260, 15.31390108383663605812660317718, 16.32862445883449798874535697422, 18.05333128127408795121761684018

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