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Properties

Label 2-2e3-8.3-c16-0-11
Degree $2$
Conductor $8$
Sign $-0.566 - 0.824i$
Analytic cond. $12.9859$
Root an. cond. $3.60360$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−242. − 81.2i)2-s − 7.66e3·3-s + (5.23e4 + 3.94e4i)4-s − 5.57e5i·5-s + (1.86e6 + 6.23e5i)6-s − 8.55e6i·7-s + (−9.49e6 − 1.38e7i)8-s + 1.57e7·9-s + (−4.52e7 + 1.35e8i)10-s + 1.51e8·11-s + (−4.01e8 − 3.02e8i)12-s − 3.64e8i·13-s + (−6.95e8 + 2.07e9i)14-s + 4.27e9i·15-s + (1.18e9 + 4.12e9i)16-s − 1.29e10·17-s + ⋯
L(s)  = 1  + (−0.948 − 0.317i)2-s − 1.16·3-s + (0.798 + 0.602i)4-s − 1.42i·5-s + (1.10 + 0.370i)6-s − 1.48i·7-s + (−0.566 − 0.824i)8-s + 0.365·9-s + (−0.452 + 1.35i)10-s + 0.704·11-s + (−0.933 − 0.703i)12-s − 0.447i·13-s + (−0.471 + 1.40i)14-s + 1.66i·15-s + (0.275 + 0.961i)16-s − 1.85·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.566 - 0.824i$
Analytic conductor: \(12.9859\)
Root analytic conductor: \(3.60360\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :8),\ -0.566 - 0.824i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.137934 + 0.262033i\)
\(L(\frac12)\) \(\approx\) \(0.137934 + 0.262033i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (242. + 81.2i)T \)
good3 \( 1 + 7.66e3T + 4.30e7T^{2} \)
5 \( 1 + 5.57e5iT - 1.52e11T^{2} \)
7 \( 1 + 8.55e6iT - 3.32e13T^{2} \)
11 \( 1 - 1.51e8T + 4.59e16T^{2} \)
13 \( 1 + 3.64e8iT - 6.65e17T^{2} \)
17 \( 1 + 1.29e10T + 4.86e19T^{2} \)
19 \( 1 - 2.11e9T + 2.88e20T^{2} \)
23 \( 1 + 7.33e10iT - 6.13e21T^{2} \)
29 \( 1 + 2.80e11iT - 2.50e23T^{2} \)
31 \( 1 - 9.63e11iT - 7.27e23T^{2} \)
37 \( 1 - 5.53e12iT - 1.23e25T^{2} \)
41 \( 1 + 1.80e11T + 6.37e25T^{2} \)
43 \( 1 - 2.29e12T + 1.36e26T^{2} \)
47 \( 1 + 1.12e13iT - 5.66e26T^{2} \)
53 \( 1 - 8.73e12iT - 3.87e27T^{2} \)
59 \( 1 - 2.07e14T + 2.15e28T^{2} \)
61 \( 1 + 8.30e12iT - 3.67e28T^{2} \)
67 \( 1 + 8.80e13T + 1.64e29T^{2} \)
71 \( 1 - 6.03e14iT - 4.16e29T^{2} \)
73 \( 1 + 1.18e15T + 6.50e29T^{2} \)
79 \( 1 + 1.98e15iT - 2.30e30T^{2} \)
83 \( 1 - 1.65e15T + 5.07e30T^{2} \)
89 \( 1 + 2.34e15T + 1.54e31T^{2} \)
97 \( 1 - 1.05e16T + 6.14e31T^{2} \)
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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

−17.03108232070328179480070231042, −16.13842369834951613396703973919, −13.08755762074579320081948686018, −11.69649653357092685848074091623, −10.42055437458436785636296603141, −8.647873923780542158218399928450, −6.68381684020973076944462578090, −4.46531352626624277674611683064, −1.11905266998439249552895021679, −0.23159504115858680143127325697, 2.28847323276074192014519034416, 5.84432332275901363857788504346, 6.81875368352948825774752551232, 9.143518985130225402425591071256, 10.99312774928433715195344429937, 11.69984657396805037328356833797, 14.73534865846755078060599766795, 15.86436198755207500061372115985, 17.55770952329000950500616774971, 18.26855084403668836184421207119

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