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Properties

Label 2-2e3-8.5-c29-0-22
Degree $2$
Conductor $8$
Sign $0.175 + 0.984i$
Analytic cond. $42.6224$
Root an. cond. $6.52858$
Motivic weight $29$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36e3 + 2.31e4i)2-s − 7.47e6i·3-s + (−5.33e8 + 6.31e7i)4-s − 2.01e10i·5-s + (1.72e11 − 1.02e10i)6-s + 1.70e12·7-s + (−2.18e12 − 1.22e13i)8-s + 1.27e13·9-s + (4.67e14 − 2.75e13i)10-s + 2.09e15i·11-s + (4.72e14 + 3.98e15i)12-s − 1.25e16i·13-s + (2.32e15 + 3.94e16i)14-s − 1.50e17·15-s + (2.80e17 − 6.73e16i)16-s + 1.08e18·17-s + ⋯
L(s)  = 1  + (0.0588 + 0.998i)2-s − 0.902i·3-s + (−0.993 + 0.117i)4-s − 1.47i·5-s + (0.901 − 0.0531i)6-s + 0.949·7-s + (−0.175 − 0.984i)8-s + 0.185·9-s + (1.47 − 0.0871i)10-s + 1.66i·11-s + (0.106 + 0.896i)12-s − 0.884i·13-s + (0.0559 + 0.947i)14-s − 1.33·15-s + (0.972 − 0.233i)16-s + 1.55·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.175 + 0.984i$
Analytic conductor: \(42.6224\)
Root analytic conductor: \(6.52858\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :29/2),\ 0.175 + 0.984i)\)

Particular Values

\(L(15)\) \(\approx\) \(1.986371374\)
\(L(\frac12)\) \(\approx\) \(1.986371374\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36e3 - 2.31e4i)T \)
good3 \( 1 + 7.47e6iT - 6.86e13T^{2} \)
5 \( 1 + 2.01e10iT - 1.86e20T^{2} \)
7 \( 1 - 1.70e12T + 3.21e24T^{2} \)
11 \( 1 - 2.09e15iT - 1.58e30T^{2} \)
13 \( 1 + 1.25e16iT - 2.01e32T^{2} \)
17 \( 1 - 1.08e18T + 4.81e35T^{2} \)
19 \( 1 + 4.89e18iT - 1.21e37T^{2} \)
23 \( 1 + 3.82e19T + 3.09e39T^{2} \)
29 \( 1 - 9.62e20iT - 2.56e42T^{2} \)
31 \( 1 + 2.34e21T + 1.77e43T^{2} \)
37 \( 1 + 7.63e22iT - 3.00e45T^{2} \)
41 \( 1 - 3.10e23T + 5.89e46T^{2} \)
43 \( 1 - 1.51e23iT - 2.34e47T^{2} \)
47 \( 1 + 1.82e24T + 3.09e48T^{2} \)
53 \( 1 + 5.09e24iT - 1.00e50T^{2} \)
59 \( 1 + 3.41e24iT - 2.26e51T^{2} \)
61 \( 1 + 7.38e25iT - 5.95e51T^{2} \)
67 \( 1 - 5.42e25iT - 9.04e52T^{2} \)
71 \( 1 - 2.54e26T + 4.85e53T^{2} \)
73 \( 1 + 4.61e25T + 1.08e54T^{2} \)
79 \( 1 + 3.25e27T + 1.07e55T^{2} \)
83 \( 1 + 4.13e27iT - 4.50e55T^{2} \)
89 \( 1 + 2.89e28T + 3.40e56T^{2} \)
97 \( 1 - 7.34e28T + 4.13e57T^{2} \)
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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

−14.55872995357221940145854843597, −12.92935737411320515447207453318, −12.40316643575507374265957278907, −9.595842070818659093822361845132, −8.099243037948641748113369850304, −7.31568000107803106092878904569, −5.37974151927747877630510319731, −4.48055336119963473496088387395, −1.60543562482617536546488696457, −0.60492518097745588728353020611, 1.46205241780639279894584602288, 3.11453178829676674380686791728, 4.01496679766995149190544820829, 5.74624731124607675232864937543, 8.057667658568281547964200709105, 9.854870962999978838119369497372, 10.78615013178183721719801829245, 11.69960533245496551085388005659, 14.05347632241797129476025150848, 14.63315666104231663867408384830

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