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Properties

Label 2-2e4-16.13-c5-0-0
Degree $2$
Conductor $16$
Sign $-0.438 - 0.898i$
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  + (−5.25 − 2.08i)2-s + (−3.18 − 3.18i)3-s + (23.3 + 21.9i)4-s + (−67.3 + 67.3i)5-s + (10.0 + 23.3i)6-s + 148. i·7-s + (−76.8 − 163. i)8-s − 222. i·9-s + (494. − 213. i)10-s + (−256. + 256. i)11-s + (−4.40 − 143. i)12-s + (−218. − 218. i)13-s + (309. − 780. i)14-s + 428.·15-s + (62.6 + 1.02e3i)16-s − 463.·17-s + ⋯
L(s)  = 1  + (−0.929 − 0.368i)2-s + (−0.203 − 0.203i)3-s + (0.728 + 0.685i)4-s + (−1.20 + 1.20i)5-s + (0.114 + 0.264i)6-s + 1.14i·7-s + (−0.424 − 0.905i)8-s − 0.916i·9-s + (1.56 − 0.676i)10-s + (−0.638 + 0.638i)11-s + (−0.00883 − 0.288i)12-s + (−0.358 − 0.358i)13-s + (0.421 − 1.06i)14-s + 0.491·15-s + (0.0612 + 0.998i)16-s − 0.388·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.438 - 0.898i$
Analytic conductor: \(2.56614\)
Root analytic conductor: \(1.60191\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5/2),\ -0.438 - 0.898i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.210995 + 0.337730i\)
\(L(\frac12)\) \(\approx\) \(0.210995 + 0.337730i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.25 + 2.08i)T \)
good3 \( 1 + (3.18 + 3.18i)T + 243iT^{2} \)
5 \( 1 + (67.3 - 67.3i)T - 3.12e3iT^{2} \)
7 \( 1 - 148. iT - 1.68e4T^{2} \)
11 \( 1 + (256. - 256. i)T - 1.61e5iT^{2} \)
13 \( 1 + (218. + 218. i)T + 3.71e5iT^{2} \)
17 \( 1 + 463.T + 1.41e6T^{2} \)
19 \( 1 + (-920. - 920. i)T + 2.47e6iT^{2} \)
23 \( 1 - 1.05e3iT - 6.43e6T^{2} \)
29 \( 1 + (-1.29e3 - 1.29e3i)T + 2.05e7iT^{2} \)
31 \( 1 - 1.00e4T + 2.86e7T^{2} \)
37 \( 1 + (9.40e3 - 9.40e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 368. iT - 1.15e8T^{2} \)
43 \( 1 + (9.16e3 - 9.16e3i)T - 1.47e8iT^{2} \)
47 \( 1 + 7.63e3T + 2.29e8T^{2} \)
53 \( 1 + (1.24e3 - 1.24e3i)T - 4.18e8iT^{2} \)
59 \( 1 + (-1.62e4 + 1.62e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (1.39e3 + 1.39e3i)T + 8.44e8iT^{2} \)
67 \( 1 + (1.74e4 + 1.74e4i)T + 1.35e9iT^{2} \)
71 \( 1 - 6.74e4iT - 1.80e9T^{2} \)
73 \( 1 + 1.95e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.39e4T + 3.07e9T^{2} \)
83 \( 1 + (-7.32e4 - 7.32e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 8.86e4iT - 5.58e9T^{2} \)
97 \( 1 + 7.36e4T + 8.58e9T^{2} \)
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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

−18.47743739538236651393092908756, −17.75346985063103747577406280728, −15.64505095177883040817335796060, −15.12454182340914749096485639956, −12.27938409902557999445249061825, −11.58502301029810087392385118216, −10.01022599308652725891341363539, −8.143753279263710150802104525325, −6.74994216799754702145016294433, −3.01657100555993255521868084764, 0.43939093951647305052290854050, 4.80753998200646947807092815008, 7.45027409225376373282120830400, 8.540704259673022286301772849491, 10.43719155874263528887439559439, 11.67565796960300762207303695214, 13.62289444803675834282333149306, 15.71346196556097066651134229779, 16.36340503006134644564080391006, 17.27902877545504515397432796494

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