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Properties

Label 2-2e3-8.3-c16-0-12
Degree $2$
Conductor $8$
Sign $-0.423 + 0.906i$
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  + (37.1 − 253. i)2-s + 1.13e4·3-s + (−6.27e4 − 1.88e4i)4-s + 8.04e3i·5-s + (4.23e5 − 2.88e6i)6-s − 7.58e6i·7-s + (−7.09e6 + 1.52e7i)8-s + 8.69e7·9-s + (2.03e6 + 2.98e5i)10-s + 1.12e8·11-s + (−7.15e8 − 2.14e8i)12-s − 1.21e9i·13-s + (−1.92e9 − 2.81e8i)14-s + 9.16e7i·15-s + (3.58e9 + 2.36e9i)16-s − 6.69e9·17-s + ⋯
L(s)  = 1  + (0.145 − 0.989i)2-s + 1.73·3-s + (−0.957 − 0.287i)4-s + 0.0205i·5-s + (0.252 − 1.71i)6-s − 1.31i·7-s + (−0.423 + 0.906i)8-s + 2.01·9-s + (0.0203 + 0.00298i)10-s + 0.526·11-s + (−1.66 − 0.498i)12-s − 1.48i·13-s + (−1.30 − 0.190i)14-s + 0.0357i·15-s + (0.835 + 0.549i)16-s − 0.960·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 + 0.906i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.423 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.423 + 0.906i$
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: \((2,\ 8,\ (\ :8),\ -0.423 + 0.906i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(1.63457 - 2.56701i\)
\(L(\frac12)\) \(\approx\) \(1.63457 - 2.56701i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-37.1 + 253. i)T \)
good3 \( 1 - 1.13e4T + 4.30e7T^{2} \)
5 \( 1 - 8.04e3iT - 1.52e11T^{2} \)
7 \( 1 + 7.58e6iT - 3.32e13T^{2} \)
11 \( 1 - 1.12e8T + 4.59e16T^{2} \)
13 \( 1 + 1.21e9iT - 6.65e17T^{2} \)
17 \( 1 + 6.69e9T + 4.86e19T^{2} \)
19 \( 1 + 4.33e9T + 2.88e20T^{2} \)
23 \( 1 - 1.36e11iT - 6.13e21T^{2} \)
29 \( 1 + 4.52e10iT - 2.50e23T^{2} \)
31 \( 1 + 1.80e11iT - 7.27e23T^{2} \)
37 \( 1 - 3.25e12iT - 1.23e25T^{2} \)
41 \( 1 - 7.38e12T + 6.37e25T^{2} \)
43 \( 1 - 1.28e13T + 1.36e26T^{2} \)
47 \( 1 - 2.62e13iT - 5.66e26T^{2} \)
53 \( 1 - 4.59e13iT - 3.87e27T^{2} \)
59 \( 1 - 7.53e13T + 2.15e28T^{2} \)
61 \( 1 - 1.29e14iT - 3.67e28T^{2} \)
67 \( 1 + 6.96e14T + 1.64e29T^{2} \)
71 \( 1 + 7.47e14iT - 4.16e29T^{2} \)
73 \( 1 - 6.29e14T + 6.50e29T^{2} \)
79 \( 1 - 5.34e13iT - 2.30e30T^{2} \)
83 \( 1 - 1.43e15T + 5.07e30T^{2} \)
89 \( 1 + 3.88e15T + 1.54e31T^{2} \)
97 \( 1 + 3.14e15T + 6.14e31T^{2} \)
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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

−17.64609607480321318093608297354, −15.11912512941904854641367729665, −13.85509631454746556497238580359, −13.03448586369572257064767906532, −10.61742185624410819267562420801, −9.258890214098545484249181322097, −7.74914698232627298806240770475, −4.12722002721307436939100577985, −2.94167002458221154575162569595, −1.17497124231622392874291644392, 2.36927336631530891690132004912, 4.26645433063817239252420709361, 6.74568242704334334498899070718, 8.648630453277458090420026685397, 9.128396989857971606255240250961, 12.63336747502579230173017425020, 14.17194699628066876292672047572, 14.91908250356781938993187817654, 16.20086087781379594028431353234, 18.38589066198568671061377096053

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