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Properties

Label 2-2e4-16.5-c9-0-2
Degree $2$
Conductor $16$
Sign $0.358 - 0.933i$
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  + (8.52 − 20.9i)2-s + (−77.8 + 77.8i)3-s + (−366. − 357. i)4-s + (206. + 206. i)5-s + (968. + 2.29e3i)6-s + 6.91e3i·7-s + (−1.06e4 + 4.64e3i)8-s + 7.56e3i·9-s + (6.08e3 − 2.56e3i)10-s + (3.14e4 + 3.14e4i)11-s + (5.63e4 − 735. i)12-s + (9.96e3 − 9.96e3i)13-s + (1.44e5 + 5.89e4i)14-s − 3.21e4·15-s + (6.84e3 + 2.62e5i)16-s − 3.78e5·17-s + ⋯
L(s)  = 1  + (0.376 − 0.926i)2-s + (−0.554 + 0.554i)3-s + (−0.716 − 0.697i)4-s + (0.147 + 0.147i)5-s + (0.304 + 0.722i)6-s + 1.08i·7-s + (−0.916 + 0.400i)8-s + 0.384i·9-s + (0.192 − 0.0812i)10-s + (0.647 + 0.647i)11-s + (0.784 − 0.0102i)12-s + (0.0967 − 0.0967i)13-s + (1.00 + 0.409i)14-s − 0.163·15-s + (0.0261 + 0.999i)16-s − 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.358 - 0.933i$
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: \((2,\ 16,\ (\ :9/2),\ 0.358 - 0.933i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.901429 + 0.619487i\)
\(L(\frac12)\) \(\approx\) \(0.901429 + 0.619487i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-8.52 + 20.9i)T \)
good3 \( 1 + (77.8 - 77.8i)T - 1.96e4iT^{2} \)
5 \( 1 + (-206. - 206. i)T + 1.95e6iT^{2} \)
7 \( 1 - 6.91e3iT - 4.03e7T^{2} \)
11 \( 1 + (-3.14e4 - 3.14e4i)T + 2.35e9iT^{2} \)
13 \( 1 + (-9.96e3 + 9.96e3i)T - 1.06e10iT^{2} \)
17 \( 1 + 3.78e5T + 1.18e11T^{2} \)
19 \( 1 + (3.90e5 - 3.90e5i)T - 3.22e11iT^{2} \)
23 \( 1 - 1.57e5iT - 1.80e12T^{2} \)
29 \( 1 + (4.13e6 - 4.13e6i)T - 1.45e13iT^{2} \)
31 \( 1 - 3.53e6T + 2.64e13T^{2} \)
37 \( 1 + (-1.43e6 - 1.43e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 1.28e7iT - 3.27e14T^{2} \)
43 \( 1 + (-2.78e7 - 2.78e7i)T + 5.02e14iT^{2} \)
47 \( 1 - 4.54e7T + 1.11e15T^{2} \)
53 \( 1 + (5.21e7 + 5.21e7i)T + 3.29e15iT^{2} \)
59 \( 1 + (-7.70e7 - 7.70e7i)T + 8.66e15iT^{2} \)
61 \( 1 + (5.07e7 - 5.07e7i)T - 1.16e16iT^{2} \)
67 \( 1 + (-1.45e7 + 1.45e7i)T - 2.72e16iT^{2} \)
71 \( 1 + 1.33e8iT - 4.58e16T^{2} \)
73 \( 1 - 1.77e8iT - 5.88e16T^{2} \)
79 \( 1 + 6.70e8T + 1.19e17T^{2} \)
83 \( 1 + (1.10e8 - 1.10e8i)T - 1.86e17iT^{2} \)
89 \( 1 + 1.10e9iT - 3.50e17T^{2} \)
97 \( 1 - 1.41e9T + 7.60e17T^{2} \)
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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

−17.41052001520865465573568750772, −15.67791942656564103807432002370, −14.45357812003461663285940848759, −12.77161897477722160571875856956, −11.56901940082439175530666375061, −10.36513367150943438894960255249, −8.959308019299900828239031744992, −5.90846444670889678864511957108, −4.41575682259419152489761622996, −2.17242860095135999508799744697, 0.52981698958370713944234757884, 4.09263846942469851278030793795, 6.11253066468562776531830914538, 7.21092227146950939935342159279, 9.060691033549199193901152487199, 11.31700120838648156778502201506, 12.92113637259798694575022911481, 13.86829488010668349751332583970, 15.39447012259813560522344486311, 17.02503285337967566285775068238

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