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Properties

Label 2-2e3-8.5-c15-0-7
Degree $2$
Conductor $8$
Sign $0.112 + 0.993i$
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  + (−153. + 96.3i)2-s − 27.2i·3-s + (1.42e4 − 2.95e4i)4-s + 2.12e5i·5-s + (2.62e3 + 4.18e3i)6-s − 3.74e6·7-s + (6.67e5 + 5.89e6i)8-s + 1.43e7·9-s + (−2.04e7 − 3.25e7i)10-s − 5.02e6i·11-s + (−8.05e5 − 3.87e5i)12-s − 3.90e8i·13-s + (5.74e8 − 3.61e8i)14-s + 5.79e6·15-s + (−6.70e8 − 8.38e8i)16-s − 2.61e8·17-s + ⋯
L(s)  = 1  + (−0.846 + 0.532i)2-s − 0.00720i·3-s + (0.433 − 0.901i)4-s + 1.21i·5-s + (0.00383 + 0.00609i)6-s − 1.72·7-s + (0.112 + 0.993i)8-s + 0.999·9-s + (−0.646 − 1.02i)10-s − 0.0777i·11-s + (−0.00648 − 0.00312i)12-s − 1.72i·13-s + (1.45 − 0.915i)14-s + 0.00875·15-s + (−0.624 − 0.781i)16-s − 0.154·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.112 + 0.993i$
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: \((2,\ 8,\ (\ :15/2),\ 0.112 + 0.993i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.306748 - 0.273967i\)
\(L(\frac12)\) \(\approx\) \(0.306748 - 0.273967i\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (153. - 96.3i)T \)
good3 \( 1 + 27.2iT - 1.43e7T^{2} \)
5 \( 1 - 2.12e5iT - 3.05e10T^{2} \)
7 \( 1 + 3.74e6T + 4.74e12T^{2} \)
11 \( 1 + 5.02e6iT - 4.17e15T^{2} \)
13 \( 1 + 3.90e8iT - 5.11e16T^{2} \)
17 \( 1 + 2.61e8T + 2.86e18T^{2} \)
19 \( 1 + 4.51e9iT - 1.51e19T^{2} \)
23 \( 1 + 4.94e9T + 2.66e20T^{2} \)
29 \( 1 + 1.04e11iT - 8.62e21T^{2} \)
31 \( 1 + 1.41e11T + 2.34e22T^{2} \)
37 \( 1 + 2.56e11iT - 3.33e23T^{2} \)
41 \( 1 - 9.25e10T + 1.55e24T^{2} \)
43 \( 1 - 1.59e12iT - 3.17e24T^{2} \)
47 \( 1 + 3.94e12T + 1.20e25T^{2} \)
53 \( 1 - 1.04e13iT - 7.31e25T^{2} \)
59 \( 1 + 1.98e13iT - 3.65e26T^{2} \)
61 \( 1 - 5.15e12iT - 6.02e26T^{2} \)
67 \( 1 + 3.73e13iT - 2.46e27T^{2} \)
71 \( 1 + 6.81e13T + 5.87e27T^{2} \)
73 \( 1 + 1.10e14T + 8.90e27T^{2} \)
79 \( 1 - 4.34e13T + 2.91e28T^{2} \)
83 \( 1 + 3.97e14iT - 6.11e28T^{2} \)
89 \( 1 + 1.70e14T + 1.74e29T^{2} \)
97 \( 1 + 7.46e14T + 6.33e29T^{2} \)
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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

−17.81784779825959444878998973315, −15.98598248817913021574016915688, −15.15243788814336098468656902635, −13.12206446686702113662318841044, −10.64275728691024160578608664186, −9.670772224765603709001296482531, −7.36259005479049375899015906371, −6.22536957924159141317456033612, −2.91556322757769772346147737762, −0.24502322543592902518104650088, 1.53055457201023939509541299948, 3.91394961111727538181404407725, 6.84711806740168294434997665683, 8.978726169576274805547785244244, 9.936156621894170707997836158277, 12.20682183052563392671142932536, 13.08608040760712557647407270226, 16.25255052392379728889985264546, 16.47904953565099172203323225893, 18.60510454125481595224375286889

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