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Properties

Label 2-10-5.3-c16-0-3
Degree $2$
Conductor $10$
Sign $0.269 + 0.962i$
Analytic cond. $16.2324$
Root an. cond. $4.02895$
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  + (−128 + 128i)2-s + (−1.32e3 − 1.32e3i)3-s − 3.27e4i·4-s + (3.90e5 − 1.60e4i)5-s + 3.38e5·6-s + (−6.21e6 + 6.21e6i)7-s + (4.19e6 + 4.19e6i)8-s − 3.95e7i·9-s + (−4.79e7 + 5.20e7i)10-s − 1.61e8·11-s + (−4.33e7 + 4.33e7i)12-s + (4.09e8 + 4.09e8i)13-s − 1.59e9i·14-s + (−5.37e8 − 4.94e8i)15-s − 1.07e9·16-s + (7.98e9 − 7.98e9i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.201 − 0.201i)3-s − 0.5i·4-s + (0.999 − 0.0410i)5-s + 0.201·6-s + (−1.07 + 1.07i)7-s + (0.250 + 0.250i)8-s − 0.918i·9-s + (−0.479 + 0.520i)10-s − 0.752·11-s + (−0.100 + 0.100i)12-s + (0.501 + 0.501i)13-s − 1.07i·14-s + (−0.209 − 0.193i)15-s − 0.250·16-s + (1.14 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 + 0.962i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.269 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.269 + 0.962i$
Analytic conductor: \(16.2324\)
Root analytic conductor: \(4.02895\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :8),\ 0.269 + 0.962i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.739996 - 0.561318i\)
\(L(\frac12)\) \(\approx\) \(0.739996 - 0.561318i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (128 - 128i)T \)
5 \( 1 + (-3.90e5 + 1.60e4i)T \)
good3 \( 1 + (1.32e3 + 1.32e3i)T + 4.30e7iT^{2} \)
7 \( 1 + (6.21e6 - 6.21e6i)T - 3.32e13iT^{2} \)
11 \( 1 + 1.61e8T + 4.59e16T^{2} \)
13 \( 1 + (-4.09e8 - 4.09e8i)T + 6.65e17iT^{2} \)
17 \( 1 + (-7.98e9 + 7.98e9i)T - 4.86e19iT^{2} \)
19 \( 1 + 2.25e10iT - 2.88e20T^{2} \)
23 \( 1 + (9.69e10 + 9.69e10i)T + 6.13e21iT^{2} \)
29 \( 1 + 4.95e11iT - 2.50e23T^{2} \)
31 \( 1 - 1.88e10T + 7.27e23T^{2} \)
37 \( 1 + (-5.70e11 + 5.70e11i)T - 1.23e25iT^{2} \)
41 \( 1 + 7.60e12T + 6.37e25T^{2} \)
43 \( 1 + (5.24e12 + 5.24e12i)T + 1.36e26iT^{2} \)
47 \( 1 + (-2.58e13 + 2.58e13i)T - 5.66e26iT^{2} \)
53 \( 1 + (-3.75e13 - 3.75e13i)T + 3.87e27iT^{2} \)
59 \( 1 - 6.84e12iT - 2.15e28T^{2} \)
61 \( 1 + 1.33e14T + 3.67e28T^{2} \)
67 \( 1 + (1.64e14 - 1.64e14i)T - 1.64e29iT^{2} \)
71 \( 1 - 2.93e14T + 4.16e29T^{2} \)
73 \( 1 + (2.10e13 + 2.10e13i)T + 6.50e29iT^{2} \)
79 \( 1 + 3.80e13iT - 2.30e30T^{2} \)
83 \( 1 + (-3.09e14 - 3.09e14i)T + 5.07e30iT^{2} \)
89 \( 1 - 5.84e15iT - 1.54e31T^{2} \)
97 \( 1 + (2.80e15 - 2.80e15i)T - 6.14e31iT^{2} \)
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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

−16.54710335657762380969542987876, −15.36947595861237136797095122573, −13.64737461194299669440027539624, −12.14423600835421278967368255822, −9.933810299133645862535374028142, −8.945409162052320779308156991396, −6.65006197101604722301927509618, −5.61847117575609099031805028210, −2.56412987221504285960909543186, −0.44416921524991227150799225485, 1.50718583547730275253680605230, 3.45457050596958245127554399182, 5.81140364160380694586820198316, 7.84250927242286572219275969924, 10.05920478511624950339394360994, 10.44557595513646876252758786719, 12.79514389094174879375864130309, 13.81639360800331007066330194651, 16.22685566393325034402924969491, 17.06493872988453266587135130031

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