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Properties

Label 2-2e3-8.5-c15-0-13
Degree $2$
Conductor $8$
Sign $-0.975 - 0.218i$
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  + (101. − 149. i)2-s − 3.58e3i·3-s + (−1.20e4 − 3.04e4i)4-s − 8.26e4i·5-s + (−5.36e5 − 3.64e5i)6-s − 1.53e5·7-s + (−5.78e6 − 1.29e6i)8-s + 1.50e6·9-s + (−1.23e7 − 8.40e6i)10-s + 2.04e7i·11-s + (−1.09e8 + 4.31e7i)12-s + 7.09e7i·13-s + (−1.56e7 + 2.30e7i)14-s − 2.96e8·15-s + (−7.83e8 + 7.34e8i)16-s − 2.71e9·17-s + ⋯
L(s)  = 1  + (0.562 − 0.826i)2-s − 0.946i·3-s + (−0.367 − 0.929i)4-s − 0.473i·5-s + (−0.782 − 0.531i)6-s − 0.0705·7-s + (−0.975 − 0.218i)8-s + 0.105·9-s + (−0.391 − 0.265i)10-s + 0.316i·11-s + (−0.879 + 0.347i)12-s + 0.313i·13-s + (−0.0396 + 0.0583i)14-s − 0.447·15-s + (−0.729 + 0.684i)16-s − 1.60·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.975 - 0.218i$
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: \((2,\ 8,\ (\ :15/2),\ -0.975 - 0.218i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.206549 + 1.86678i\)
\(L(\frac12)\) \(\approx\) \(0.206549 + 1.86678i\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-101. + 149. i)T \)
good3 \( 1 + 3.58e3iT - 1.43e7T^{2} \)
5 \( 1 + 8.26e4iT - 3.05e10T^{2} \)
7 \( 1 + 1.53e5T + 4.74e12T^{2} \)
11 \( 1 - 2.04e7iT - 4.17e15T^{2} \)
13 \( 1 - 7.09e7iT - 5.11e16T^{2} \)
17 \( 1 + 2.71e9T + 2.86e18T^{2} \)
19 \( 1 + 4.09e9iT - 1.51e19T^{2} \)
23 \( 1 - 2.81e10T + 2.66e20T^{2} \)
29 \( 1 + 1.54e11iT - 8.62e21T^{2} \)
31 \( 1 + 7.02e10T + 2.34e22T^{2} \)
37 \( 1 + 4.56e11iT - 3.33e23T^{2} \)
41 \( 1 + 4.22e11T + 1.55e24T^{2} \)
43 \( 1 - 9.06e11iT - 3.17e24T^{2} \)
47 \( 1 - 3.38e12T + 1.20e25T^{2} \)
53 \( 1 - 5.59e11iT - 7.31e25T^{2} \)
59 \( 1 - 2.62e13iT - 3.65e26T^{2} \)
61 \( 1 + 3.59e13iT - 6.02e26T^{2} \)
67 \( 1 - 1.80e13iT - 2.46e27T^{2} \)
71 \( 1 - 1.14e14T + 5.87e27T^{2} \)
73 \( 1 + 5.99e12T + 8.90e27T^{2} \)
79 \( 1 - 1.01e14T + 2.91e28T^{2} \)
83 \( 1 - 1.34e14iT - 6.11e28T^{2} \)
89 \( 1 - 8.80e13T + 1.74e29T^{2} \)
97 \( 1 - 1.21e15T + 6.33e29T^{2} \)
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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

−17.60353652921931530625891157819, −15.34146587459185822670907313366, −13.47235082007949381094049811068, −12.71034147302906094220982726074, −11.18880723438809676457921012863, −9.134933325778866823752726971887, −6.74580748410731836541544162221, −4.61838618882448653815558711890, −2.26060425701340803843345779057, −0.71952787391488251631604046979, 3.36664471257542095012347101100, 4.94456496568379657868790086306, 6.85893511656504111032990865240, 8.916834394822868079858869139938, 10.82930068829028668552269449763, 12.95151387173362662535629781220, 14.62511866228630650037801312898, 15.61972436436048412693106368547, 16.81259553727745889735590732241, 18.42407079560323290400040198725

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