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Properties

Label 2-2e3-8.5-c29-0-15
Degree $2$
Conductor $8$
Sign $0.751 + 0.659i$
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  + (1.60e4 − 1.67e4i)2-s + 3.75e6i·3-s + (−2.33e7 − 5.36e8i)4-s − 2.08e9i·5-s + (6.27e10 + 6.01e10i)6-s + 1.73e12·7-s + (−9.35e12 − 8.20e12i)8-s + 5.45e13·9-s + (−3.48e13 − 3.33e13i)10-s + 1.58e15i·11-s + (2.01e15 − 8.74e13i)12-s + 1.00e16i·13-s + (2.77e16 − 2.90e16i)14-s + 7.81e15·15-s + (−2.87e17 + 2.50e16i)16-s − 1.73e17·17-s + ⋯
L(s)  = 1  + (0.691 − 0.722i)2-s + 0.452i·3-s + (−0.0434 − 0.999i)4-s − 0.152i·5-s + (0.327 + 0.313i)6-s + 0.966·7-s + (−0.751 − 0.659i)8-s + 0.794·9-s + (−0.110 − 0.105i)10-s + 1.25i·11-s + (0.452 − 0.0196i)12-s + 0.710i·13-s + (0.668 − 0.698i)14-s + 0.0691·15-s + (−0.996 + 0.0867i)16-s − 0.250·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.751 + 0.659i$
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: \((2,\ 8,\ (\ :29/2),\ 0.751 + 0.659i)\)

Particular Values

\(L(15)\) \(\approx\) \(3.659789199\)
\(L(\frac12)\) \(\approx\) \(3.659789199\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.60e4 + 1.67e4i)T \)
good3 \( 1 - 3.75e6iT - 6.86e13T^{2} \)
5 \( 1 + 2.08e9iT - 1.86e20T^{2} \)
7 \( 1 - 1.73e12T + 3.21e24T^{2} \)
11 \( 1 - 1.58e15iT - 1.58e30T^{2} \)
13 \( 1 - 1.00e16iT - 2.01e32T^{2} \)
17 \( 1 + 1.73e17T + 4.81e35T^{2} \)
19 \( 1 + 2.89e18iT - 1.21e37T^{2} \)
23 \( 1 - 4.98e19T + 3.09e39T^{2} \)
29 \( 1 + 1.27e21iT - 2.56e42T^{2} \)
31 \( 1 - 5.57e21T + 1.77e43T^{2} \)
37 \( 1 - 6.56e21iT - 3.00e45T^{2} \)
41 \( 1 - 2.69e23T + 5.89e46T^{2} \)
43 \( 1 + 5.10e23iT - 2.34e47T^{2} \)
47 \( 1 - 2.24e24T + 3.09e48T^{2} \)
53 \( 1 + 1.08e25iT - 1.00e50T^{2} \)
59 \( 1 + 4.11e25iT - 2.26e51T^{2} \)
61 \( 1 - 6.80e25iT - 5.95e51T^{2} \)
67 \( 1 - 1.27e26iT - 9.04e52T^{2} \)
71 \( 1 + 4.60e26T + 4.85e53T^{2} \)
73 \( 1 + 1.90e27T + 1.08e54T^{2} \)
79 \( 1 - 2.53e26T + 1.07e55T^{2} \)
83 \( 1 - 2.57e27iT - 4.50e55T^{2} \)
89 \( 1 - 1.95e28T + 3.40e56T^{2} \)
97 \( 1 + 3.15e28T + 4.13e57T^{2} \)
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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

−14.63120058610515043479920987560, −13.12090519978007666254778128366, −11.78323713337045128513250966888, −10.49156459104664498441040077370, −9.203461331228896766103055878638, −6.97391153094584841885453137727, −4.88500568151723415756873068404, −4.29258780563631926552460957942, −2.32629950940397679074384877319, −1.11585779765989845507498343556, 1.08337817149803623021569971149, 2.95804932209141032697940305640, 4.57027854291523979243865864059, 5.98050590015521382142421158476, 7.41858351510619612849950818308, 8.516452136448626801009754140298, 10.94959930761153610703791519397, 12.46714080019203838326572332492, 13.68039017981591980419510092910, 14.85424913628783776171548742985

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