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Properties

Label 2-2e4-16.3-c10-0-17
Degree $2$
Conductor $16$
Sign $-0.275 + 0.961i$
Analytic cond. $10.1657$
Root an. cond. $3.18837$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (26.0 − 18.5i)2-s + (164. − 164. i)3-s + (337. − 966. i)4-s + (897. − 897. i)5-s + (1.24e3 − 7.35e3i)6-s + 594.·7-s + (−9.09e3 − 3.14e4i)8-s + 4.67e3i·9-s + (6.79e3 − 4.00e4i)10-s + (2.97e3 + 2.97e3i)11-s + (−1.03e5 − 2.15e5i)12-s + (−3.27e4 − 3.27e4i)13-s + (1.55e4 − 1.10e4i)14-s − 2.95e5i·15-s + (−8.20e5 − 6.52e5i)16-s − 5.91e5·17-s + ⋯
L(s)  = 1  + (0.815 − 0.578i)2-s + (0.678 − 0.678i)3-s + (0.329 − 0.944i)4-s + (0.287 − 0.287i)5-s + (0.160 − 0.946i)6-s + 0.0353·7-s + (−0.277 − 0.960i)8-s + 0.0791i·9-s + (0.0679 − 0.400i)10-s + (0.0184 + 0.0184i)11-s + (−0.416 − 0.864i)12-s + (−0.0883 − 0.0883i)13-s + (0.0288 − 0.0204i)14-s − 0.389i·15-s + (−0.782 − 0.622i)16-s − 0.416·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.275 + 0.961i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ -0.275 + 0.961i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.04048 - 2.70817i\)
\(L(\frac12)\) \(\approx\) \(2.04048 - 2.70817i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-26.0 + 18.5i)T \)
good3 \( 1 + (-164. + 164. i)T - 5.90e4iT^{2} \)
5 \( 1 + (-897. + 897. i)T - 9.76e6iT^{2} \)
7 \( 1 - 594.T + 2.82e8T^{2} \)
11 \( 1 + (-2.97e3 - 2.97e3i)T + 2.59e10iT^{2} \)
13 \( 1 + (3.27e4 + 3.27e4i)T + 1.37e11iT^{2} \)
17 \( 1 + 5.91e5T + 2.01e12T^{2} \)
19 \( 1 + (-1.87e6 + 1.87e6i)T - 6.13e12iT^{2} \)
23 \( 1 - 3.55e6T + 4.14e13T^{2} \)
29 \( 1 + (-1.92e7 - 1.92e7i)T + 4.20e14iT^{2} \)
31 \( 1 - 5.18e7iT - 8.19e14T^{2} \)
37 \( 1 + (-6.64e7 + 6.64e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 1.67e8iT - 1.34e16T^{2} \)
43 \( 1 + (3.64e7 + 3.64e7i)T + 2.16e16iT^{2} \)
47 \( 1 - 2.54e8iT - 5.25e16T^{2} \)
53 \( 1 + (-4.68e7 + 4.68e7i)T - 1.74e17iT^{2} \)
59 \( 1 + (-1.66e8 - 1.66e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (5.92e8 + 5.92e8i)T + 7.13e17iT^{2} \)
67 \( 1 + (1.32e9 - 1.32e9i)T - 1.82e18iT^{2} \)
71 \( 1 - 2.48e8T + 3.25e18T^{2} \)
73 \( 1 + 1.98e9iT - 4.29e18T^{2} \)
79 \( 1 + 7.20e8iT - 9.46e18T^{2} \)
83 \( 1 + (3.72e9 - 3.72e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 1.51e9iT - 3.11e19T^{2} \)
97 \( 1 + 7.26e9T + 7.37e19T^{2} \)
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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

−15.95760539065171480589049990824, −14.42974379386906618866245192917, −13.47727851827270460794789351276, −12.49301880948674940043919060926, −10.84942180588304220290400303513, −9.058435766603788309299018500922, −7.05814320736357117553115307698, −5.07431479700081865162576210770, −2.90821587863378686640806760984, −1.37279887619561078464118915738, 2.83811093232617356631495521048, 4.36342504356451529684252257116, 6.28636923402123218403240449033, 8.131873254401459150898447108980, 9.765962996191542104870183258523, 11.72023085959048901713958966212, 13.40655565543676705399560476610, 14.53778285735935212776732869522, 15.40179056096194789018412123000, 16.66773803981765779319345870909

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