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Properties

Label 2-2e3-8.3-c16-0-5
Degree $2$
Conductor $8$
Sign $0.964 + 0.264i$
Analytic cond. $12.9859$
Root an. cond. $3.60360$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−107. − 232. i)2-s + 693.·3-s + (−4.23e4 + 5.00e4i)4-s − 3.33e5i·5-s + (−7.46e4 − 1.60e5i)6-s + 1.01e7i·7-s + (1.61e7 + 4.43e6i)8-s − 4.25e7·9-s + (−7.73e7 + 3.58e7i)10-s + 3.53e8·11-s + (−2.93e7 + 3.46e7i)12-s − 4.13e8i·13-s + (2.35e9 − 1.09e9i)14-s − 2.30e8i·15-s + (−7.12e8 − 4.23e9i)16-s + 7.22e9·17-s + ⋯
L(s)  = 1  + (−0.420 − 0.907i)2-s + 0.105·3-s + (−0.645 + 0.763i)4-s − 0.852i·5-s + (−0.0444 − 0.0958i)6-s + 1.75i·7-s + (0.964 + 0.264i)8-s − 0.988·9-s + (−0.773 + 0.358i)10-s + 1.65·11-s + (−0.0682 + 0.0806i)12-s − 0.506i·13-s + (1.59 − 0.739i)14-s − 0.0900i·15-s + (−0.165 − 0.986i)16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.964 + 0.264i$
Analytic conductor: \(12.9859\)
Root analytic conductor: \(3.60360\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :8),\ 0.964 + 0.264i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(1.37501 - 0.185198i\)
\(L(\frac12)\) \(\approx\) \(1.37501 - 0.185198i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (107. + 232. i)T \)
good3 \( 1 - 693.T + 4.30e7T^{2} \)
5 \( 1 + 3.33e5iT - 1.52e11T^{2} \)
7 \( 1 - 1.01e7iT - 3.32e13T^{2} \)
11 \( 1 - 3.53e8T + 4.59e16T^{2} \)
13 \( 1 + 4.13e8iT - 6.65e17T^{2} \)
17 \( 1 - 7.22e9T + 4.86e19T^{2} \)
19 \( 1 - 1.17e10T + 2.88e20T^{2} \)
23 \( 1 - 3.06e10iT - 6.13e21T^{2} \)
29 \( 1 - 4.93e11iT - 2.50e23T^{2} \)
31 \( 1 - 8.90e11iT - 7.27e23T^{2} \)
37 \( 1 - 2.63e12iT - 1.23e25T^{2} \)
41 \( 1 - 1.13e13T + 6.37e25T^{2} \)
43 \( 1 + 6.39e11T + 1.36e26T^{2} \)
47 \( 1 + 9.74e12iT - 5.66e26T^{2} \)
53 \( 1 + 3.29e13iT - 3.87e27T^{2} \)
59 \( 1 + 7.37e13T + 2.15e28T^{2} \)
61 \( 1 - 3.01e14iT - 3.67e28T^{2} \)
67 \( 1 - 6.34e14T + 1.64e29T^{2} \)
71 \( 1 - 3.99e12iT - 4.16e29T^{2} \)
73 \( 1 + 7.97e14T + 6.50e29T^{2} \)
79 \( 1 + 9.10e14iT - 2.30e30T^{2} \)
83 \( 1 - 7.04e14T + 5.07e30T^{2} \)
89 \( 1 + 1.16e15T + 1.54e31T^{2} \)
97 \( 1 + 3.41e15T + 6.14e31T^{2} \)
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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

−17.81774395866576792243438194006, −16.50026610747508433806853379912, −14.44171760384715399810190727008, −12.42528428727354869938261018541, −11.69104088322318012776738774883, −9.294775551575173411711052339752, −8.536481487129183713332045062116, −5.38653380030244506637117964229, −3.09065411220274613576488858269, −1.26839501227598652429847617434, 0.842620766893523494102672025170, 3.92448794065837812990122188108, 6.37578247815655809153238164908, 7.59373667847457654408699269868, 9.577676994220679805641426636965, 11.11086125484475536996415172605, 14.06342473322024135567999782708, 14.42944033042282287852281177915, 16.64314449716451430334209963761, 17.34585679796634302791140883461

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