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Properties

Label 2-2e4-16.5-c7-0-0
Degree $2$
Conductor $16$
Sign $-0.239 + 0.970i$
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  + (−3.93 + 10.6i)2-s + (−35.6 + 35.6i)3-s + (−97.0 − 83.4i)4-s + (50.2 + 50.2i)5-s + (−238. − 518. i)6-s − 699. i·7-s + (1.26e3 − 701. i)8-s − 356. i·9-s + (−729. + 335. i)10-s + (−5.55e3 − 5.55e3i)11-s + (6.43e3 − 487. i)12-s + (−5.45e3 + 5.45e3i)13-s + (7.41e3 + 2.74e3i)14-s − 3.58e3·15-s + (2.46e3 + 1.61e4i)16-s − 1.86e3·17-s + ⋯
L(s)  = 1  + (−0.347 + 0.937i)2-s + (−0.762 + 0.762i)3-s + (−0.758 − 0.651i)4-s + (0.179 + 0.179i)5-s + (−0.450 − 0.980i)6-s − 0.770i·7-s + (0.874 − 0.484i)8-s − 0.163i·9-s + (−0.230 + 0.105i)10-s + (−1.25 − 1.25i)11-s + (1.07 − 0.0814i)12-s + (−0.688 + 0.688i)13-s + (0.722 + 0.267i)14-s − 0.273·15-s + (0.150 + 0.988i)16-s − 0.0922·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.239 + 0.970i$
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: \((2,\ 16,\ (\ :7/2),\ -0.239 + 0.970i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0334620 - 0.0427061i\)
\(L(\frac12)\) \(\approx\) \(0.0334620 - 0.0427061i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.93 - 10.6i)T \)
good3 \( 1 + (35.6 - 35.6i)T - 2.18e3iT^{2} \)
5 \( 1 + (-50.2 - 50.2i)T + 7.81e4iT^{2} \)
7 \( 1 + 699. iT - 8.23e5T^{2} \)
11 \( 1 + (5.55e3 + 5.55e3i)T + 1.94e7iT^{2} \)
13 \( 1 + (5.45e3 - 5.45e3i)T - 6.27e7iT^{2} \)
17 \( 1 + 1.86e3T + 4.10e8T^{2} \)
19 \( 1 + (2.69e4 - 2.69e4i)T - 8.93e8iT^{2} \)
23 \( 1 + 2.18e4iT - 3.40e9T^{2} \)
29 \( 1 + (3.62e4 - 3.62e4i)T - 1.72e10iT^{2} \)
31 \( 1 + 8.09e4T + 2.75e10T^{2} \)
37 \( 1 + (-3.21e5 - 3.21e5i)T + 9.49e10iT^{2} \)
41 \( 1 - 4.20e5iT - 1.94e11T^{2} \)
43 \( 1 + (-3.56e5 - 3.56e5i)T + 2.71e11iT^{2} \)
47 \( 1 + 1.25e6T + 5.06e11T^{2} \)
53 \( 1 + (1.29e6 + 1.29e6i)T + 1.17e12iT^{2} \)
59 \( 1 + (-1.51e6 - 1.51e6i)T + 2.48e12iT^{2} \)
61 \( 1 + (2.17e6 - 2.17e6i)T - 3.14e12iT^{2} \)
67 \( 1 + (3.82e5 - 3.82e5i)T - 6.06e12iT^{2} \)
71 \( 1 + 1.32e6iT - 9.09e12T^{2} \)
73 \( 1 + 3.62e6iT - 1.10e13T^{2} \)
79 \( 1 - 3.87e6T + 1.92e13T^{2} \)
83 \( 1 + (-1.05e6 + 1.05e6i)T - 2.71e13iT^{2} \)
89 \( 1 + 2.24e6iT - 4.42e13T^{2} \)
97 \( 1 + 4.58e6T + 8.07e13T^{2} \)
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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

−18.08970307936333070833167706389, −16.57445505038196441721911785855, −16.38617837344771183227543950011, −14.68998249073206032836314381761, −13.37192370205952628323151260581, −10.90925146573014905499072723377, −10.01641499353131037789979004910, −8.031197678428101977531339861314, −6.14148704504856556289500503724, −4.62565201524194671890295453043, 0.03968918017116347008489607632, 2.21470803758626220637265557144, 5.22069391693140078145942524786, 7.53204148298071431043297820588, 9.441007425245714473480256777324, 11.01715060101608752196847807230, 12.49711389592505258392982020327, 12.92492258672839884691612993878, 15.24803801613049215105228089372, 17.31476983725780658721476213017

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