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Properties

Label 2-2e3-8.5-c15-0-2
Degree $2$
Conductor $8$
Sign $-0.579 - 0.814i$
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  + (134. − 120. i)2-s + 6.53e3i·3-s + (3.64e3 − 3.25e4i)4-s + 1.16e5i·5-s + (7.88e5 + 8.82e5i)6-s − 2.94e6·7-s + (−3.43e6 − 4.83e6i)8-s − 2.83e7·9-s + (1.40e7 + 1.57e7i)10-s + 5.06e7i·11-s + (2.12e8 + 2.38e7i)12-s + 7.14e7i·13-s + (−3.97e8 + 3.55e8i)14-s − 7.61e8·15-s + (−1.04e9 − 2.37e8i)16-s + 1.88e9·17-s + ⋯
L(s)  = 1  + (0.745 − 0.666i)2-s + 1.72i·3-s + (0.111 − 0.993i)4-s + 0.666i·5-s + (1.15 + 1.28i)6-s − 1.35·7-s + (−0.579 − 0.814i)8-s − 1.97·9-s + (0.444 + 0.497i)10-s + 0.784i·11-s + (1.71 + 0.191i)12-s + 0.315i·13-s + (−1.00 + 0.900i)14-s − 1.15·15-s + (−0.975 − 0.221i)16-s + 1.11·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.579 - 0.814i$
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: \((2,\ 8,\ (\ :15/2),\ -0.579 - 0.814i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.725790 + 1.40676i\)
\(L(\frac12)\) \(\approx\) \(0.725790 + 1.40676i\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-134. + 120. i)T \)
good3 \( 1 - 6.53e3iT - 1.43e7T^{2} \)
5 \( 1 - 1.16e5iT - 3.05e10T^{2} \)
7 \( 1 + 2.94e6T + 4.74e12T^{2} \)
11 \( 1 - 5.06e7iT - 4.17e15T^{2} \)
13 \( 1 - 7.14e7iT - 5.11e16T^{2} \)
17 \( 1 - 1.88e9T + 2.86e18T^{2} \)
19 \( 1 - 5.68e9iT - 1.51e19T^{2} \)
23 \( 1 + 1.88e10T + 2.66e20T^{2} \)
29 \( 1 + 7.71e10iT - 8.62e21T^{2} \)
31 \( 1 - 2.10e11T + 2.34e22T^{2} \)
37 \( 1 - 1.49e11iT - 3.33e23T^{2} \)
41 \( 1 + 5.40e11T + 1.55e24T^{2} \)
43 \( 1 - 1.85e12iT - 3.17e24T^{2} \)
47 \( 1 - 5.09e12T + 1.20e25T^{2} \)
53 \( 1 - 1.42e12iT - 7.31e25T^{2} \)
59 \( 1 - 1.44e13iT - 3.65e26T^{2} \)
61 \( 1 - 4.18e12iT - 6.02e26T^{2} \)
67 \( 1 + 5.42e13iT - 2.46e27T^{2} \)
71 \( 1 + 3.53e12T + 5.87e27T^{2} \)
73 \( 1 + 6.90e13T + 8.90e27T^{2} \)
79 \( 1 + 2.39e14T + 2.91e28T^{2} \)
83 \( 1 - 1.38e14iT - 6.11e28T^{2} \)
89 \( 1 + 1.79e14T + 1.74e29T^{2} \)
97 \( 1 - 1.74e14T + 6.33e29T^{2} \)
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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

−18.93674005107045786520763107206, −16.43741128372070843730005614881, −15.32031270817059491251415719865, −14.19214505695896308606617027689, −12.09347707474101394700051866824, −10.24504260402255729775719159867, −9.797961760328962090646191698438, −6.00883775378012532308731348064, −4.13777517808186742392538665525, −3.00523991847508938435665195205, 0.54192841125828026459723361639, 2.94661819094486222538300831854, 5.79810520885911252980690869109, 7.01416337273945513079788062430, 8.537283507426254390214721007899, 12.08292581364544170148282622984, 12.96919650247752249850816884663, 13.86989069350257376095395639636, 16.05547044910582825935704213240, 17.29399315158341377664743511671

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