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Properties

Label 2-2e4-16.5-c9-0-11
Degree $2$
Conductor $16$
Sign $0.448 + 0.893i$
Analytic cond. $8.24057$
Root an. cond. $2.87063$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (18.5 − 12.9i)2-s + (62.9 − 62.9i)3-s + (178. − 479. i)4-s + (1.82e3 + 1.82e3i)5-s + (357. − 1.98e3i)6-s − 7.94e3i·7-s + (−2.87e3 − 1.12e4i)8-s + 1.17e4i·9-s + (5.76e4 + 1.03e4i)10-s + (−3.45e4 − 3.45e4i)11-s + (−1.89e4 − 4.14e4i)12-s + (−3.72e3 + 3.72e3i)13-s + (−1.02e5 − 1.47e5i)14-s + 2.30e5·15-s + (−1.98e5 − 1.71e5i)16-s + 4.38e5·17-s + ⋯
L(s)  = 1  + (0.821 − 0.570i)2-s + (0.448 − 0.448i)3-s + (0.348 − 0.937i)4-s + (1.30 + 1.30i)5-s + (0.112 − 0.624i)6-s − 1.25i·7-s + (−0.248 − 0.968i)8-s + 0.597i·9-s + (1.82 + 0.328i)10-s + (−0.710 − 0.710i)11-s + (−0.263 − 0.577i)12-s + (−0.0361 + 0.0361i)13-s + (−0.713 − 1.02i)14-s + 1.17·15-s + (−0.756 − 0.654i)16-s + 1.27·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.448 + 0.893i$
Analytic conductor: \(8.24057\)
Root analytic conductor: \(2.87063\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :9/2),\ 0.448 + 0.893i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.88787 - 1.78191i\)
\(L(\frac12)\) \(\approx\) \(2.88787 - 1.78191i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-18.5 + 12.9i)T \)
good3 \( 1 + (-62.9 + 62.9i)T - 1.96e4iT^{2} \)
5 \( 1 + (-1.82e3 - 1.82e3i)T + 1.95e6iT^{2} \)
7 \( 1 + 7.94e3iT - 4.03e7T^{2} \)
11 \( 1 + (3.45e4 + 3.45e4i)T + 2.35e9iT^{2} \)
13 \( 1 + (3.72e3 - 3.72e3i)T - 1.06e10iT^{2} \)
17 \( 1 - 4.38e5T + 1.18e11T^{2} \)
19 \( 1 + (6.17e5 - 6.17e5i)T - 3.22e11iT^{2} \)
23 \( 1 - 5.08e5iT - 1.80e12T^{2} \)
29 \( 1 + (2.04e6 - 2.04e6i)T - 1.45e13iT^{2} \)
31 \( 1 + 1.29e6T + 2.64e13T^{2} \)
37 \( 1 + (-2.27e6 - 2.27e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 1.77e7iT - 3.27e14T^{2} \)
43 \( 1 + (2.37e6 + 2.37e6i)T + 5.02e14iT^{2} \)
47 \( 1 + 1.63e6T + 1.11e15T^{2} \)
53 \( 1 + (7.92e6 + 7.92e6i)T + 3.29e15iT^{2} \)
59 \( 1 + (-7.11e6 - 7.11e6i)T + 8.66e15iT^{2} \)
61 \( 1 + (5.30e7 - 5.30e7i)T - 1.16e16iT^{2} \)
67 \( 1 + (1.09e7 - 1.09e7i)T - 2.72e16iT^{2} \)
71 \( 1 - 9.17e7iT - 4.58e16T^{2} \)
73 \( 1 + 3.75e8iT - 5.88e16T^{2} \)
79 \( 1 + 3.90e8T + 1.19e17T^{2} \)
83 \( 1 + (-1.22e8 + 1.22e8i)T - 1.86e17iT^{2} \)
89 \( 1 + 5.82e8iT - 3.50e17T^{2} \)
97 \( 1 - 1.03e9T + 7.60e17T^{2} \)
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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

−16.73910199946135482877459135832, −14.55263201495780937085340714331, −13.92045953061259272667024720943, −13.09117616738313565966494287738, −10.76077864119230754591343922831, −10.20625393754398173059373487746, −7.31844308776878404545396068224, −5.78012188054900269378638791099, −3.24316375259831116599051822491, −1.77293535119288653059155107024, 2.41199078126278090508886132405, 4.81869662163207130412891960649, 5.96819676370401064570221512957, 8.512941233549057220160270153051, 9.570645093309946235534436859352, 12.30892529927367221855018865698, 13.06109608585140089957102168980, 14.61681316061293599651485513027, 15.61935502875914777198408879611, 16.93340131085631565367210474561

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