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Properties

Label 2-2e3-8.3-c10-0-7
Degree $2$
Conductor $8$
Sign $0.684 + 0.728i$
Analytic cond. $5.08285$
Root an. cond. $2.25451$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (22.8 − 22.3i)2-s + 352.·3-s + (21.3 − 1.02e3i)4-s + 3.77e3i·5-s + (8.04e3 − 7.88e3i)6-s − 1.56e4i·7-s + (−2.24e4 − 2.38e4i)8-s + 6.49e4·9-s + (8.44e4 + 8.62e4i)10-s − 1.15e5·11-s + (7.50e3 − 3.60e5i)12-s + 5.46e5i·13-s + (−3.49e5 − 3.57e5i)14-s + 1.32e6i·15-s + (−1.04e6 − 4.36e4i)16-s − 1.50e6·17-s + ⋯
L(s)  = 1  + (0.714 − 0.699i)2-s + 1.44·3-s + (0.0208 − 0.999i)4-s + 1.20i·5-s + (1.03 − 1.01i)6-s − 0.929i·7-s + (−0.684 − 0.728i)8-s + 1.09·9-s + (0.844 + 0.862i)10-s − 0.715·11-s + (0.0301 − 1.44i)12-s + 1.47i·13-s + (−0.650 − 0.663i)14-s + 1.74i·15-s + (−0.999 − 0.0416i)16-s − 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.684 + 0.728i$
Analytic conductor: \(5.08285\)
Root analytic conductor: \(2.25451\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :5),\ 0.684 + 0.728i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.78904 - 1.20664i\)
\(L(\frac12)\) \(\approx\) \(2.78904 - 1.20664i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-22.8 + 22.3i)T \)
good3 \( 1 - 352.T + 5.90e4T^{2} \)
5 \( 1 - 3.77e3iT - 9.76e6T^{2} \)
7 \( 1 + 1.56e4iT - 2.82e8T^{2} \)
11 \( 1 + 1.15e5T + 2.59e10T^{2} \)
13 \( 1 - 5.46e5iT - 1.37e11T^{2} \)
17 \( 1 + 1.50e6T + 2.01e12T^{2} \)
19 \( 1 - 3.45e6T + 6.13e12T^{2} \)
23 \( 1 - 3.90e6iT - 4.14e13T^{2} \)
29 \( 1 + 1.57e7iT - 4.20e14T^{2} \)
31 \( 1 + 1.36e7iT - 8.19e14T^{2} \)
37 \( 1 + 7.96e7iT - 4.80e15T^{2} \)
41 \( 1 + 1.01e7T + 1.34e16T^{2} \)
43 \( 1 - 4.09e7T + 2.16e16T^{2} \)
47 \( 1 + 2.36e8iT - 5.25e16T^{2} \)
53 \( 1 - 3.09e8iT - 1.74e17T^{2} \)
59 \( 1 + 6.26e7T + 5.11e17T^{2} \)
61 \( 1 + 1.02e9iT - 7.13e17T^{2} \)
67 \( 1 - 6.25e8T + 1.82e18T^{2} \)
71 \( 1 - 2.05e9iT - 3.25e18T^{2} \)
73 \( 1 - 2.44e9T + 4.29e18T^{2} \)
79 \( 1 - 3.04e9iT - 9.46e18T^{2} \)
83 \( 1 - 2.75e9T + 1.55e19T^{2} \)
89 \( 1 - 2.51e8T + 3.11e19T^{2} \)
97 \( 1 + 1.56e10T + 7.37e19T^{2} \)
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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

−19.51867184187687845903717071027, −18.45195919446228200977878561288, −15.47678272731714208625602723762, −14.14321705556692567120942349719, −13.60576946764701976697146798727, −11.14522924196690554300869532634, −9.605941534019428139921687422008, −7.13880003209401711245068878383, −3.81354918231039245425319484914, −2.34563759868110248128275819356, 2.87954643047333169984286693771, 5.15701318819513387851534183383, 8.012064800682004586603855909817, 8.954944392736945258929670685861, 12.52956518757466387483808519062, 13.49988774598160904225892745487, 15.10743918181038319011651998452, 15.99109180338736116317382429146, 18.01220257692599238221241025363, 20.14303781218562231456728006870

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