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Properties

Label 2-2e4-16.11-c10-0-13
Degree $2$
Conductor $16$
Sign $0.755 + 0.655i$
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  + (30.4 − 9.83i)2-s + (−61.4 − 61.4i)3-s + (830. − 598. i)4-s + (2.59e3 + 2.59e3i)5-s + (−2.47e3 − 1.26e3i)6-s + 6.39e3·7-s + (1.94e4 − 2.64e4i)8-s − 5.14e4i·9-s + (1.04e5 + 5.35e4i)10-s + (1.56e5 − 1.56e5i)11-s + (−8.78e4 − 1.42e4i)12-s + (−2.57e5 + 2.57e5i)13-s + (1.94e5 − 6.28e4i)14-s − 3.19e5i·15-s + (3.31e5 − 9.94e5i)16-s + 3.54e5·17-s + ⋯
L(s)  = 1  + (0.951 − 0.307i)2-s + (−0.252 − 0.252i)3-s + (0.811 − 0.584i)4-s + (0.830 + 0.830i)5-s + (−0.318 − 0.162i)6-s + 0.380·7-s + (0.592 − 0.805i)8-s − 0.872i·9-s + (1.04 + 0.535i)10-s + (0.973 − 0.973i)11-s + (−0.353 − 0.0572i)12-s + (−0.692 + 0.692i)13-s + (0.362 − 0.116i)14-s − 0.420i·15-s + (0.316 − 0.948i)16-s + 0.249·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.755 + 0.655i$
Analytic conductor: \(10.1657\)
Root analytic conductor: \(3.18837\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5),\ 0.755 + 0.655i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(3.12640 - 1.16670i\)
\(L(\frac12)\) \(\approx\) \(3.12640 - 1.16670i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-30.4 + 9.83i)T \)
good3 \( 1 + (61.4 + 61.4i)T + 5.90e4iT^{2} \)
5 \( 1 + (-2.59e3 - 2.59e3i)T + 9.76e6iT^{2} \)
7 \( 1 - 6.39e3T + 2.82e8T^{2} \)
11 \( 1 + (-1.56e5 + 1.56e5i)T - 2.59e10iT^{2} \)
13 \( 1 + (2.57e5 - 2.57e5i)T - 1.37e11iT^{2} \)
17 \( 1 - 3.54e5T + 2.01e12T^{2} \)
19 \( 1 + (-2.12e6 - 2.12e6i)T + 6.13e12iT^{2} \)
23 \( 1 + 6.22e6T + 4.14e13T^{2} \)
29 \( 1 + (2.63e7 - 2.63e7i)T - 4.20e14iT^{2} \)
31 \( 1 - 2.50e7iT - 8.19e14T^{2} \)
37 \( 1 + (2.20e7 + 2.20e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 4.56e7iT - 1.34e16T^{2} \)
43 \( 1 + (-8.56e7 + 8.56e7i)T - 2.16e16iT^{2} \)
47 \( 1 - 1.47e8iT - 5.25e16T^{2} \)
53 \( 1 + (3.63e8 + 3.63e8i)T + 1.74e17iT^{2} \)
59 \( 1 + (-5.92e8 + 5.92e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (5.38e8 - 5.38e8i)T - 7.13e17iT^{2} \)
67 \( 1 + (-1.19e9 - 1.19e9i)T + 1.82e18iT^{2} \)
71 \( 1 + 1.49e8T + 3.25e18T^{2} \)
73 \( 1 + 2.05e9iT - 4.29e18T^{2} \)
79 \( 1 - 3.02e9iT - 9.46e18T^{2} \)
83 \( 1 + (2.60e9 + 2.60e9i)T + 1.55e19iT^{2} \)
89 \( 1 - 8.66e9iT - 3.11e19T^{2} \)
97 \( 1 + 1.42e10T + 7.37e19T^{2} \)
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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

−16.43385171196462367985871118248, −14.47541309257549587727923869795, −14.15472169930435932260192605077, −12.30815201601067799509984310863, −11.22911589443702588469205695701, −9.673405223349759045041496728244, −6.85744920256126645922722324189, −5.75361311400345131048528464826, −3.49174747584905177260311537967, −1.57403857694797492619365550505, 1.97477862069739341667813374307, 4.55305019229967126059795898654, 5.63291578556897760482188799492, 7.63858431969887814426092669280, 9.731153309507342270127816264971, 11.57607136028616573265681947509, 12.89780074896249469208929740921, 14.03178535698825267537352705831, 15.39665286143491423454643986103, 16.84639338340155783199786072436

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