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Properties

Label 2-2e4-16.3-c10-0-2
Degree $2$
Conductor $16$
Sign $-0.908 + 0.418i$
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  + (12.5 + 29.4i)2-s + (44.0 − 44.0i)3-s + (−709. + 738. i)4-s + (−126. + 126. i)5-s + (1.84e3 + 745. i)6-s − 1.95e4·7-s + (−3.06e4 − 1.16e4i)8-s + 5.51e4i·9-s + (−5.30e3 − 2.13e3i)10-s + (−3.55e4 − 3.55e4i)11-s + (1.24e3 + 6.38e4i)12-s + (−3.16e5 − 3.16e5i)13-s + (−2.45e5 − 5.76e5i)14-s + 1.11e4i·15-s + (−4.09e4 − 1.04e6i)16-s + 3.57e5·17-s + ⋯
L(s)  = 1  + (0.391 + 0.920i)2-s + (0.181 − 0.181i)3-s + (−0.693 + 0.720i)4-s + (−0.0404 + 0.0404i)5-s + (0.237 + 0.0958i)6-s − 1.16·7-s + (−0.934 − 0.355i)8-s + 0.934i·9-s + (−0.0530 − 0.0213i)10-s + (−0.221 − 0.221i)11-s + (0.00500 + 0.256i)12-s + (−0.853 − 0.853i)13-s + (−0.456 − 1.07i)14-s + 0.0146i·15-s + (−0.0390 − 0.999i)16-s + 0.252·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.908 + 0.418i$
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: \((2,\ 16,\ (\ :5),\ -0.908 + 0.418i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.150856 - 0.687984i\)
\(L(\frac12)\) \(\approx\) \(0.150856 - 0.687984i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-12.5 - 29.4i)T \)
good3 \( 1 + (-44.0 + 44.0i)T - 5.90e4iT^{2} \)
5 \( 1 + (126. - 126. i)T - 9.76e6iT^{2} \)
7 \( 1 + 1.95e4T + 2.82e8T^{2} \)
11 \( 1 + (3.55e4 + 3.55e4i)T + 2.59e10iT^{2} \)
13 \( 1 + (3.16e5 + 3.16e5i)T + 1.37e11iT^{2} \)
17 \( 1 - 3.57e5T + 2.01e12T^{2} \)
19 \( 1 + (1.92e6 - 1.92e6i)T - 6.13e12iT^{2} \)
23 \( 1 - 1.53e6T + 4.14e13T^{2} \)
29 \( 1 + (-1.81e7 - 1.81e7i)T + 4.20e14iT^{2} \)
31 \( 1 - 2.17e7iT - 8.19e14T^{2} \)
37 \( 1 + (2.50e7 - 2.50e7i)T - 4.80e15iT^{2} \)
41 \( 1 - 1.41e8iT - 1.34e16T^{2} \)
43 \( 1 + (9.95e7 + 9.95e7i)T + 2.16e16iT^{2} \)
47 \( 1 + 3.56e8iT - 5.25e16T^{2} \)
53 \( 1 + (-1.39e8 + 1.39e8i)T - 1.74e17iT^{2} \)
59 \( 1 + (3.54e8 + 3.54e8i)T + 5.11e17iT^{2} \)
61 \( 1 + (-6.20e8 - 6.20e8i)T + 7.13e17iT^{2} \)
67 \( 1 + (-1.28e9 + 1.28e9i)T - 1.82e18iT^{2} \)
71 \( 1 + 2.69e9T + 3.25e18T^{2} \)
73 \( 1 - 2.03e9iT - 4.29e18T^{2} \)
79 \( 1 + 1.53e9iT - 9.46e18T^{2} \)
83 \( 1 + (3.27e9 - 3.27e9i)T - 1.55e19iT^{2} \)
89 \( 1 - 4.89e9iT - 3.11e19T^{2} \)
97 \( 1 - 3.33e8T + 7.37e19T^{2} \)
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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

−17.02703951578587085648359050190, −16.11108350214355497748094261563, −14.81773655970700950310122015113, −13.41267139135049953277456933014, −12.51074517842400886496852906701, −10.12869454092168378084834654513, −8.340312206737098137081162063728, −6.93925822597529693947529282358, −5.27270647976501135572666122327, −3.14486459221040191572335319079, 0.27165896488241902089025889127, 2.67282552933144026268052568994, 4.27663349616546603161377682039, 6.42273784209093843781377248051, 9.132133235897924625390331513309, 10.10042141190640929555740266978, 11.91046079029903441347857826893, 12.92243643229794034676822501406, 14.37926856318899037341488603019, 15.65226112801370297837299164489

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