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Properties

Label 2-2e3-8.5-c15-0-5
Degree $2$
Conductor $8$
Sign $0.454 - 0.890i$
Analytic cond. $11.4154$
Root an. cond. $3.37868$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−28.3 + 178. i)2-s − 2.38e3i·3-s + (−3.11e4 − 1.01e4i)4-s + 9.94e3i·5-s + (4.25e5 + 6.75e4i)6-s + 1.24e6·7-s + (2.69e6 − 5.28e6i)8-s + 8.68e6·9-s + (−1.77e6 − 2.82e5i)10-s + 6.55e7i·11-s + (−2.41e7 + 7.41e7i)12-s + 1.91e8i·13-s + (−3.52e7 + 2.22e8i)14-s + 2.36e7·15-s + (8.68e8 + 6.31e8i)16-s + 1.98e9·17-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s − 0.628i·3-s + (−0.950 − 0.309i)4-s + 0.0569i·5-s + (0.620 + 0.0984i)6-s + 0.570·7-s + (0.454 − 0.890i)8-s + 0.605·9-s + (−0.0562 − 0.00891i)10-s + 1.01i·11-s + (−0.194 + 0.597i)12-s + 0.848i·13-s + (−0.0894 + 0.563i)14-s + 0.0357·15-s + (0.808 + 0.588i)16-s + 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.454 - 0.890i$
Analytic conductor: \(11.4154\)
Root analytic conductor: \(3.37868\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :15/2),\ 0.454 - 0.890i)\)

Particular Values

\(L(8)\) \(\approx\) \(1.41087 + 0.863878i\)
\(L(\frac12)\) \(\approx\) \(1.41087 + 0.863878i\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (28.3 - 178. i)T \)
good3 \( 1 + 2.38e3iT - 1.43e7T^{2} \)
5 \( 1 - 9.94e3iT - 3.05e10T^{2} \)
7 \( 1 - 1.24e6T + 4.74e12T^{2} \)
11 \( 1 - 6.55e7iT - 4.17e15T^{2} \)
13 \( 1 - 1.91e8iT - 5.11e16T^{2} \)
17 \( 1 - 1.98e9T + 2.86e18T^{2} \)
19 \( 1 - 8.13e8iT - 1.51e19T^{2} \)
23 \( 1 - 8.30e8T + 2.66e20T^{2} \)
29 \( 1 + 5.26e10iT - 8.62e21T^{2} \)
31 \( 1 - 2.31e11T + 2.34e22T^{2} \)
37 \( 1 - 5.88e11iT - 3.33e23T^{2} \)
41 \( 1 - 2.88e11T + 1.55e24T^{2} \)
43 \( 1 + 1.77e12iT - 3.17e24T^{2} \)
47 \( 1 + 6.45e12T + 1.20e25T^{2} \)
53 \( 1 - 1.55e13iT - 7.31e25T^{2} \)
59 \( 1 + 1.28e13iT - 3.65e26T^{2} \)
61 \( 1 - 3.34e13iT - 6.02e26T^{2} \)
67 \( 1 + 8.71e13iT - 2.46e27T^{2} \)
71 \( 1 - 1.73e13T + 5.87e27T^{2} \)
73 \( 1 - 1.25e14T + 8.90e27T^{2} \)
79 \( 1 + 4.05e12T + 2.91e28T^{2} \)
83 \( 1 - 2.09e14iT - 6.11e28T^{2} \)
89 \( 1 + 3.43e14T + 1.74e29T^{2} \)
97 \( 1 + 7.44e14T + 6.33e29T^{2} \)
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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

−18.11827186631497217145323117930, −16.82946296242294582725472381359, −15.21034915004176946737706068536, −13.90713931475711586640727820164, −12.33889836438257030252531966362, −9.864132088927522991075711430540, −7.945056036137496546848898758525, −6.69511159713131215329291428294, −4.63794636947557754880443891325, −1.34431366198644523186534183977, 1.01684451869694760017490180377, 3.28388718137026319372159849752, 5.00723757980019379597841479663, 8.242344297235532002118318935582, 9.938469568599178590433359176799, 11.15762692568822830632298298501, 12.84173627790003751411771327405, 14.50036076481464803034311256159, 16.36849399881183898615239755702, 17.96100487937671495746602561671

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