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Properties

Label 2-2e3-8.5-c19-0-9
Degree $2$
Conductor $8$
Sign $0.696 - 0.717i$
Analytic cond. $18.3053$
Root an. cond. $4.27847$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (698. − 190. i)2-s + 2.93e4i·3-s + (4.51e5 − 2.66e5i)4-s + 7.95e5i·5-s + (5.59e6 + 2.04e7i)6-s + 3.26e7·7-s + (2.64e8 − 2.72e8i)8-s + 3.03e8·9-s + (1.51e8 + 5.55e8i)10-s + 7.73e9i·11-s + (7.81e9 + 1.32e10i)12-s + 4.36e10i·13-s + (2.28e10 − 6.23e9i)14-s − 2.33e10·15-s + (1.32e11 − 2.40e11i)16-s + 6.06e10·17-s + ⋯
L(s)  = 1  + (0.964 − 0.263i)2-s + 0.859i·3-s + (0.861 − 0.508i)4-s + 0.182i·5-s + (0.226 + 0.829i)6-s + 0.305·7-s + (0.696 − 0.717i)8-s + 0.260·9-s + (0.0479 + 0.175i)10-s + 0.989i·11-s + (0.437 + 0.740i)12-s + 1.14i·13-s + (0.295 − 0.0806i)14-s − 0.156·15-s + (0.482 − 0.875i)16-s + 0.123·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.696 - 0.717i$
Analytic conductor: \(18.3053\)
Root analytic conductor: \(4.27847\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :19/2),\ 0.696 - 0.717i)\)

Particular Values

\(L(10)\) \(\approx\) \(3.43403 + 1.45222i\)
\(L(\frac12)\) \(\approx\) \(3.43403 + 1.45222i\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-698. + 190. i)T \)
good3 \( 1 - 2.93e4iT - 1.16e9T^{2} \)
5 \( 1 - 7.95e5iT - 1.90e13T^{2} \)
7 \( 1 - 3.26e7T + 1.13e16T^{2} \)
11 \( 1 - 7.73e9iT - 6.11e19T^{2} \)
13 \( 1 - 4.36e10iT - 1.46e21T^{2} \)
17 \( 1 - 6.06e10T + 2.39e23T^{2} \)
19 \( 1 - 9.91e11iT - 1.97e24T^{2} \)
23 \( 1 - 6.78e12T + 7.46e25T^{2} \)
29 \( 1 + 7.31e13iT - 6.10e27T^{2} \)
31 \( 1 + 1.38e14T + 2.16e28T^{2} \)
37 \( 1 + 9.51e14iT - 6.24e29T^{2} \)
41 \( 1 + 1.79e15T + 4.39e30T^{2} \)
43 \( 1 + 4.29e15iT - 1.08e31T^{2} \)
47 \( 1 + 2.72e15T + 5.88e31T^{2} \)
53 \( 1 - 1.42e16iT - 5.77e32T^{2} \)
59 \( 1 + 5.38e16iT - 4.42e33T^{2} \)
61 \( 1 + 1.47e17iT - 8.34e33T^{2} \)
67 \( 1 - 3.11e17iT - 4.95e34T^{2} \)
71 \( 1 - 6.27e17T + 1.49e35T^{2} \)
73 \( 1 - 1.71e17T + 2.53e35T^{2} \)
79 \( 1 - 1.66e18T + 1.13e36T^{2} \)
83 \( 1 + 1.40e18iT - 2.90e36T^{2} \)
89 \( 1 + 5.88e18T + 1.09e37T^{2} \)
97 \( 1 + 1.38e19T + 5.60e37T^{2} \)
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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

−16.64996711581818495075285025959, −15.33417591832285903594192142910, −14.31983381727466931636411016803, −12.55192301849493516039088596080, −11.01436781107068323140634691423, −9.674769549347795941787653280815, −6.98888292012354900356707457717, −4.97797739427469191824326858529, −3.83296280119343161619385661741, −1.85174608823908056205057583229, 1.17089537917863706019680589696, 3.05181946663306000720622542082, 5.13059576339252900238363034404, 6.74809633525455883959757366971, 8.163441467583939336856041741287, 11.01791875374336437561048308440, 12.60106785742242105409932141799, 13.49237472082967779414124665829, 15.03947924610113410938327729237, 16.55643505196951225496224122765

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