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Properties

Label 2-2e3-8.5-c13-0-6
Degree $2$
Conductor $8$
Sign $-0.908 + 0.417i$
Analytic cond. $8.57847$
Root an. cond. $2.92890$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−56 − 71.1i)2-s + 2.29e3i·3-s + (−1.92e3 + 7.96e3i)4-s − 2.25e4i·5-s + (1.63e5 − 1.28e5i)6-s − 1.75e5·7-s + (6.73e5 − 3.09e5i)8-s − 3.66e6·9-s + (−1.60e6 + 1.26e6i)10-s + 2.63e6i·11-s + (−1.82e7 − 4.40e6i)12-s − 3.10e7i·13-s + (9.84e6 + 1.25e7i)14-s + 5.17e7·15-s + (−5.97e7 − 3.05e7i)16-s − 1.33e8·17-s + ⋯
L(s)  = 1  + (−0.618 − 0.785i)2-s + 1.81i·3-s + (−0.234 + 0.972i)4-s − 0.646i·5-s + (1.42 − 1.12i)6-s − 0.564·7-s + (0.908 − 0.417i)8-s − 2.29·9-s + (−0.507 + 0.399i)10-s + 0.448i·11-s + (−1.76 − 0.425i)12-s − 1.78i·13-s + (0.349 + 0.443i)14-s + 1.17·15-s + (−0.890 − 0.455i)16-s − 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.908 + 0.417i$
Analytic conductor: \(8.57847\)
Root analytic conductor: \(2.92890\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 8,\ (\ :13/2),\ -0.908 + 0.417i)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (56 + 71.1i)T \)
good3 \( 1 - 2.29e3iT - 1.59e6T^{2} \)
5 \( 1 + 2.25e4iT - 1.22e9T^{2} \)
7 \( 1 + 1.75e5T + 9.68e10T^{2} \)
11 \( 1 - 2.63e6iT - 3.45e13T^{2} \)
13 \( 1 + 3.10e7iT - 3.02e14T^{2} \)
17 \( 1 + 1.33e8T + 9.90e15T^{2} \)
19 \( 1 + 3.47e7iT - 4.20e16T^{2} \)
23 \( 1 + 3.55e7T + 5.04e17T^{2} \)
29 \( 1 - 1.58e9iT - 1.02e19T^{2} \)
31 \( 1 + 5.76e9T + 2.44e19T^{2} \)
37 \( 1 - 1.31e10iT - 2.43e20T^{2} \)
41 \( 1 + 2.35e10T + 9.25e20T^{2} \)
43 \( 1 - 1.45e10iT - 1.71e21T^{2} \)
47 \( 1 + 6.81e10T + 5.46e21T^{2} \)
53 \( 1 - 1.66e11iT - 2.60e22T^{2} \)
59 \( 1 + 1.27e11iT - 1.04e23T^{2} \)
61 \( 1 + 4.24e11iT - 1.61e23T^{2} \)
67 \( 1 - 3.76e11iT - 5.48e23T^{2} \)
71 \( 1 + 1.30e12T + 1.16e24T^{2} \)
73 \( 1 - 4.78e11T + 1.67e24T^{2} \)
79 \( 1 + 3.64e11T + 4.66e24T^{2} \)
83 \( 1 - 8.72e11iT - 8.87e24T^{2} \)
89 \( 1 + 1.02e11T + 2.19e25T^{2} \)
97 \( 1 + 6.15e12T + 6.73e25T^{2} \)
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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

−17.65978435455203681643641082054, −16.43926881792246532000931553469, −15.33668974110885581988608251445, −12.87621409939849304338204397150, −10.93507854705696446556708009729, −9.825415577534778085788850462259, −8.645963212763147773751643674215, −4.81004787998134418614049204748, −3.19818504863189000629106719430, 0, 1.93038818587889765107238007347, 6.35634786931397683112681367751, 7.09964062377997907717772034749, 8.846768203395262984579593176778, 11.33823241226993906262910766164, 13.32932010916070356905866113244, 14.42478382858237728831045644709, 16.53113069873533837985788904387, 17.96913617909271316794322981383

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