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Properties

Label 2-2e3-8.5-c29-0-13
Degree $2$
Conductor $8$
Sign $0.939 + 0.341i$
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  + (−2.30e4 − 2.68e3i)2-s + 1.92e6i·3-s + (5.22e8 + 1.23e8i)4-s + 2.40e10i·5-s + (5.17e9 − 4.44e10i)6-s − 2.26e12·7-s + (−1.16e13 − 4.24e12i)8-s + 6.49e13·9-s + (6.44e13 − 5.52e14i)10-s − 1.44e15i·11-s + (−2.38e14 + 1.00e15i)12-s − 1.68e15i·13-s + (5.20e16 + 6.07e15i)14-s − 4.63e16·15-s + (2.57e17 + 1.29e17i)16-s − 8.60e17·17-s + ⋯
L(s)  = 1  + (−0.993 − 0.115i)2-s + 0.232i·3-s + (0.973 + 0.230i)4-s + 1.76i·5-s + (0.0269 − 0.231i)6-s − 1.26·7-s + (−0.939 − 0.341i)8-s + 0.945·9-s + (0.203 − 1.74i)10-s − 1.15i·11-s + (−0.0535 + 0.226i)12-s − 0.118i·13-s + (1.25 + 0.146i)14-s − 0.409·15-s + (0.894 + 0.447i)16-s − 1.23·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.939 + 0.341i$
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: \((2,\ 8,\ (\ :29/2),\ 0.939 + 0.341i)\)

Particular Values

\(L(15)\) \(\approx\) \(0.8151794573\)
\(L(\frac12)\) \(\approx\) \(0.8151794573\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.30e4 + 2.68e3i)T \)
good3 \( 1 - 1.92e6iT - 6.86e13T^{2} \)
5 \( 1 - 2.40e10iT - 1.86e20T^{2} \)
7 \( 1 + 2.26e12T + 3.21e24T^{2} \)
11 \( 1 + 1.44e15iT - 1.58e30T^{2} \)
13 \( 1 + 1.68e15iT - 2.01e32T^{2} \)
17 \( 1 + 8.60e17T + 4.81e35T^{2} \)
19 \( 1 + 6.15e18iT - 1.21e37T^{2} \)
23 \( 1 - 2.99e19T + 3.09e39T^{2} \)
29 \( 1 - 1.64e21iT - 2.56e42T^{2} \)
31 \( 1 - 2.37e21T + 1.77e43T^{2} \)
37 \( 1 - 2.96e22iT - 3.00e45T^{2} \)
41 \( 1 - 2.80e23T + 5.89e46T^{2} \)
43 \( 1 + 5.14e23iT - 2.34e47T^{2} \)
47 \( 1 - 4.44e23T + 3.09e48T^{2} \)
53 \( 1 + 1.85e25iT - 1.00e50T^{2} \)
59 \( 1 - 6.66e25iT - 2.26e51T^{2} \)
61 \( 1 + 4.35e25iT - 5.95e51T^{2} \)
67 \( 1 + 8.97e25iT - 9.04e52T^{2} \)
71 \( 1 - 1.17e26T + 4.85e53T^{2} \)
73 \( 1 + 8.98e26T + 1.08e54T^{2} \)
79 \( 1 - 2.09e27T + 1.07e55T^{2} \)
83 \( 1 - 2.66e27iT - 4.50e55T^{2} \)
89 \( 1 - 2.29e28T + 3.40e56T^{2} \)
97 \( 1 - 6.39e28T + 4.13e57T^{2} \)
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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

−15.25126905165594815611477310095, −13.28257376668541455617221480385, −11.19082534283021182582150201105, −10.39823898563506868169109441095, −9.128127560261413540392255011114, −7.05144370342070525983896596511, −6.48420671447132996258750424109, −3.40790019740797500157950806134, −2.55226839387264735538604878688, −0.41886001200443959445773300313, 0.862352435567200284811564386943, 1.96868755164358078898084477180, 4.32609382258691849305233686412, 6.20429793500670285818088259656, 7.67456776352960356819391242948, 9.167977499201536980806912934179, 9.954600013816565025937707169426, 12.26108969407100525314806707871, 12.95815214936743837105449812232, 15.59307211555801978871495730503

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