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Properties

Label 2-2e4-16.5-c5-0-1
Degree $2$
Conductor $16$
Sign $-0.918 - 0.395i$
Analytic cond. $2.56614$
Root an. cond. $1.60191$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0198 + 5.65i)2-s + (−8.39 + 8.39i)3-s + (−31.9 − 0.224i)4-s + (−8.40 − 8.40i)5-s + (−47.3 − 47.6i)6-s + 149. i·7-s + (1.90 − 181. i)8-s + 102. i·9-s + (47.7 − 47.3i)10-s + (349. + 349. i)11-s + (270. − 266. i)12-s + (450. − 450. i)13-s + (−844. − 2.96i)14-s + 141.·15-s + (1.02e3 + 14.3i)16-s − 1.29e3·17-s + ⋯
L(s)  = 1  + (−0.00351 + 0.999i)2-s + (−0.538 + 0.538i)3-s + (−0.999 − 0.00702i)4-s + (−0.150 − 0.150i)5-s + (−0.536 − 0.540i)6-s + 1.15i·7-s + (0.0105 − 0.999i)8-s + 0.420i·9-s + (0.150 − 0.149i)10-s + (0.871 + 0.871i)11-s + (0.542 − 0.534i)12-s + (0.738 − 0.738i)13-s + (−1.15 − 0.00404i)14-s + 0.161·15-s + (0.999 + 0.0140i)16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.918 - 0.395i$
Analytic conductor: \(2.56614\)
Root analytic conductor: \(1.60191\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5/2),\ -0.918 - 0.395i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.186000 + 0.901919i\)
\(L(\frac12)\) \(\approx\) \(0.186000 + 0.901919i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0198 - 5.65i)T \)
good3 \( 1 + (8.39 - 8.39i)T - 243iT^{2} \)
5 \( 1 + (8.40 + 8.40i)T + 3.12e3iT^{2} \)
7 \( 1 - 149. iT - 1.68e4T^{2} \)
11 \( 1 + (-349. - 349. i)T + 1.61e5iT^{2} \)
13 \( 1 + (-450. + 450. i)T - 3.71e5iT^{2} \)
17 \( 1 + 1.29e3T + 1.41e6T^{2} \)
19 \( 1 + (-116. + 116. i)T - 2.47e6iT^{2} \)
23 \( 1 - 3.52e3iT - 6.43e6T^{2} \)
29 \( 1 + (-3.45e3 + 3.45e3i)T - 2.05e7iT^{2} \)
31 \( 1 - 5.05e3T + 2.86e7T^{2} \)
37 \( 1 + (-7.89e3 - 7.89e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.84e4iT - 1.15e8T^{2} \)
43 \( 1 + (1.12e4 + 1.12e4i)T + 1.47e8iT^{2} \)
47 \( 1 + 2.44e3T + 2.29e8T^{2} \)
53 \( 1 + (-7.56e3 - 7.56e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (4.63e3 + 4.63e3i)T + 7.14e8iT^{2} \)
61 \( 1 + (-2.47e4 + 2.47e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (8.65e3 - 8.65e3i)T - 1.35e9iT^{2} \)
71 \( 1 + 1.57e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.53e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.74e4T + 3.07e9T^{2} \)
83 \( 1 + (2.43e4 - 2.43e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 7.88e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.55e5T + 8.58e9T^{2} \)
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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

−18.17120911660428825255397101957, −17.21117250349398962711356101490, −15.79786431067695394974076272201, −15.21741880865068222859822171945, −13.41067210051227022962962462780, −11.77333139445987970771972663499, −9.790767256488700177473937564503, −8.308334197602335895302394570573, −6.19158420586800257421751519564, −4.67053664961069596365237551043, 0.898454342830445475550668983926, 3.95412033268393818441108161327, 6.60679155723562280793850620296, 8.885262401604982104240862977373, 10.77810215547873922139759514702, 11.70891954466616056664426143127, 13.19317168171277658254890837310, 14.28974838368759446021351267233, 16.64789869943353952717805668549, 17.73986567253343053119758138838

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