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Properties

Label 2-2e3-8.5-c19-0-17
Degree $2$
Conductor $8$
Sign $-0.440 - 0.897i$
Analytic cond. $18.3053$
Root an. cond. $4.27847$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (564. − 452. i)2-s − 5.04e4i·3-s + (1.14e5 − 5.11e5i)4-s + 7.26e6i·5-s + (−2.28e7 − 2.84e7i)6-s − 1.80e8·7-s + (−1.67e8 − 3.40e8i)8-s − 1.38e9·9-s + (3.28e9 + 4.10e9i)10-s + 1.74e9i·11-s + (−2.58e10 − 5.75e9i)12-s + 9.15e9i·13-s + (−1.02e11 + 8.19e10i)14-s + 3.66e11·15-s + (−2.48e11 − 1.16e11i)16-s − 2.51e11·17-s + ⋯
L(s)  = 1  + (0.780 − 0.625i)2-s − 1.47i·3-s + (0.217 − 0.976i)4-s + 1.66i·5-s + (−0.925 − 1.15i)6-s − 1.69·7-s + (−0.440 − 0.897i)8-s − 1.18·9-s + (1.04 + 1.29i)10-s + 0.223i·11-s + (−1.44 − 0.321i)12-s + 0.239i·13-s + (−1.32 + 1.06i)14-s + 2.46·15-s + (−0.905 − 0.424i)16-s − 0.514·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.440 - 0.897i$
Analytic conductor: \(18.3053\)
Root analytic conductor: \(4.27847\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :19/2),\ -0.440 - 0.897i)\)

Particular Values

\(L(10)\) \(\approx\) \(0.315465 + 0.506374i\)
\(L(\frac12)\) \(\approx\) \(0.315465 + 0.506374i\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-564. + 452. i)T \)
good3 \( 1 + 5.04e4iT - 1.16e9T^{2} \)
5 \( 1 - 7.26e6iT - 1.90e13T^{2} \)
7 \( 1 + 1.80e8T + 1.13e16T^{2} \)
11 \( 1 - 1.74e9iT - 6.11e19T^{2} \)
13 \( 1 - 9.15e9iT - 1.46e21T^{2} \)
17 \( 1 + 2.51e11T + 2.39e23T^{2} \)
19 \( 1 + 1.49e12iT - 1.97e24T^{2} \)
23 \( 1 + 4.50e12T + 7.46e25T^{2} \)
29 \( 1 - 3.77e12iT - 6.10e27T^{2} \)
31 \( 1 - 1.61e14T + 2.16e28T^{2} \)
37 \( 1 - 4.58e13iT - 6.24e29T^{2} \)
41 \( 1 + 3.30e15T + 4.39e30T^{2} \)
43 \( 1 + 7.16e14iT - 1.08e31T^{2} \)
47 \( 1 + 9.61e15T + 5.88e31T^{2} \)
53 \( 1 + 2.87e16iT - 5.77e32T^{2} \)
59 \( 1 + 8.42e16iT - 4.42e33T^{2} \)
61 \( 1 - 3.85e16iT - 8.34e33T^{2} \)
67 \( 1 - 3.60e16iT - 4.95e34T^{2} \)
71 \( 1 + 8.39e16T + 1.49e35T^{2} \)
73 \( 1 - 1.83e14T + 2.53e35T^{2} \)
79 \( 1 - 1.00e18T + 1.13e36T^{2} \)
83 \( 1 - 1.33e18iT - 2.90e36T^{2} \)
89 \( 1 + 4.03e18T + 1.09e37T^{2} \)
97 \( 1 - 6.57e18T + 5.60e37T^{2} \)
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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

−15.45169914520973652218844240165, −13.88351288117272109138450472817, −12.99592600757690499777399445122, −11.61026403262032826040247729269, −10.02003464351836960419835104973, −6.82155267291466347723436271281, −6.46630035094955359970668581403, −3.23647478322546103715717056289, −2.24207716039105267176272163400, −0.15850747222322163598465458566, 3.45213717485655731975946873263, 4.59565096295353086229317582821, 5.93340855926478710327025434777, 8.608055511508073105083978559096, 9.837339334926724416030726418872, 12.25201770360627609552907192627, 13.42932994656456753519936081284, 15.47120025804113693228605978061, 16.25319614665928360176651979395, 16.84752622187531768493383007353

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