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Properties

Label 2-2e4-16.5-c7-0-4
Degree $2$
Conductor $16$
Sign $0.847 - 0.531i$
Analytic cond. $4.99816$
Root an. cond. $2.23565$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.66 − 10.9i)2-s + (−21.1 + 21.1i)3-s + (−113. + 58.7i)4-s + (328. + 328. i)5-s + (288. + 175. i)6-s − 874. i·7-s + (949. + 1.09e3i)8-s + 1.29e3i·9-s + (2.73e3 − 4.49e3i)10-s + (4.82e3 + 4.82e3i)11-s + (1.16e3 − 3.63e3i)12-s + (−7.51e3 + 7.51e3i)13-s + (−9.61e3 + 2.33e3i)14-s − 1.38e4·15-s + (9.49e3 − 1.33e4i)16-s − 7.42e3·17-s + ⋯
L(s)  = 1  + (−0.235 − 0.971i)2-s + (−0.451 + 0.451i)3-s + (−0.888 + 0.458i)4-s + (1.17 + 1.17i)5-s + (0.545 + 0.332i)6-s − 0.963i·7-s + (0.655 + 0.755i)8-s + 0.592i·9-s + (0.865 − 1.42i)10-s + (1.09 + 1.09i)11-s + (0.194 − 0.608i)12-s + (−0.948 + 0.948i)13-s + (−0.936 + 0.227i)14-s − 1.06·15-s + (0.579 − 0.815i)16-s − 0.366·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.847 - 0.531i$
Analytic conductor: \(4.99816\)
Root analytic conductor: \(2.23565\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :7/2),\ 0.847 - 0.531i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.16573 + 0.335379i\)
\(L(\frac12)\) \(\approx\) \(1.16573 + 0.335379i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.66 + 10.9i)T \)
good3 \( 1 + (21.1 - 21.1i)T - 2.18e3iT^{2} \)
5 \( 1 + (-328. - 328. i)T + 7.81e4iT^{2} \)
7 \( 1 + 874. iT - 8.23e5T^{2} \)
11 \( 1 + (-4.82e3 - 4.82e3i)T + 1.94e7iT^{2} \)
13 \( 1 + (7.51e3 - 7.51e3i)T - 6.27e7iT^{2} \)
17 \( 1 + 7.42e3T + 4.10e8T^{2} \)
19 \( 1 + (-3.62e3 + 3.62e3i)T - 8.93e8iT^{2} \)
23 \( 1 + 1.85e4iT - 3.40e9T^{2} \)
29 \( 1 + (-8.19e4 + 8.19e4i)T - 1.72e10iT^{2} \)
31 \( 1 - 1.06e5T + 2.75e10T^{2} \)
37 \( 1 + (-1.68e5 - 1.68e5i)T + 9.49e10iT^{2} \)
41 \( 1 + 8.07e4iT - 1.94e11T^{2} \)
43 \( 1 + (4.71e5 + 4.71e5i)T + 2.71e11iT^{2} \)
47 \( 1 + 1.24e5T + 5.06e11T^{2} \)
53 \( 1 + (-5.33e5 - 5.33e5i)T + 1.17e12iT^{2} \)
59 \( 1 + (7.60e4 + 7.60e4i)T + 2.48e12iT^{2} \)
61 \( 1 + (-2.46e6 + 2.46e6i)T - 3.14e12iT^{2} \)
67 \( 1 + (9.03e5 - 9.03e5i)T - 6.06e12iT^{2} \)
71 \( 1 - 2.53e6iT - 9.09e12T^{2} \)
73 \( 1 - 4.78e5iT - 1.10e13T^{2} \)
79 \( 1 + 2.01e6T + 1.92e13T^{2} \)
83 \( 1 + (-5.67e6 + 5.67e6i)T - 2.71e13iT^{2} \)
89 \( 1 + 8.04e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.06e6T + 8.07e13T^{2} \)
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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

−17.45929969357150712912784145893, −17.03013635712144436255404207065, −14.47654987292195071882434508398, −13.59531648666858848226783138936, −11.65313294107975807671467930108, −10.34570533926955101565859116266, −9.708879771145607942894361448480, −6.94778630689375411155359572650, −4.47327149659316622733156861723, −2.10865549821987110474544028015, 0.946975864822838241039504471812, 5.34701431163650117094458435612, 6.29154565523366902501142818196, 8.621978422001301800722395550192, 9.578792770511459616707511337301, 12.19358874445936176957680458025, 13.33439998314703879307534387712, 14.78612029126658621013642036125, 16.38961481549111925341591542795, 17.37860938214225409417786022964

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