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Properties

Label 2-10-5.2-c18-0-8
Degree $2$
Conductor $10$
Sign $-0.622 + 0.782i$
Analytic cond. $20.5386$
Root an. cond. $4.53195$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (256 + 256i)2-s + (2.59e4 − 2.59e4i)3-s + 1.31e5i·4-s + (−1.76e6 − 8.32e5i)5-s + 1.33e7·6-s + (−2.67e7 − 2.67e7i)7-s + (−3.35e7 + 3.35e7i)8-s − 9.62e8i·9-s + (−2.39e8 − 6.65e8i)10-s − 7.63e8·11-s + (3.40e9 + 3.40e9i)12-s + (−1.08e10 + 1.08e10i)13-s − 1.36e10i·14-s + (−6.75e10 + 2.42e10i)15-s − 1.71e10·16-s + (4.48e10 + 4.48e10i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (1.31 − 1.31i)3-s + 0.5i·4-s + (−0.904 − 0.426i)5-s + 1.31·6-s + (−0.662 − 0.662i)7-s + (−0.250 + 0.250i)8-s − 2.48i·9-s + (−0.239 − 0.665i)10-s − 0.323·11-s + (0.659 + 0.659i)12-s + (−1.02 + 1.02i)13-s − 0.662i·14-s + (−1.75 + 0.631i)15-s − 0.250·16-s + (0.378 + 0.378i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.622 + 0.782i$
Analytic conductor: \(20.5386\)
Root analytic conductor: \(4.53195\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :9),\ -0.622 + 0.782i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.944650 - 1.95871i\)
\(L(\frac12)\) \(\approx\) \(0.944650 - 1.95871i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-256 - 256i)T \)
5 \( 1 + (1.76e6 + 8.32e5i)T \)
good3 \( 1 + (-2.59e4 + 2.59e4i)T - 3.87e8iT^{2} \)
7 \( 1 + (2.67e7 + 2.67e7i)T + 1.62e15iT^{2} \)
11 \( 1 + 7.63e8T + 5.55e18T^{2} \)
13 \( 1 + (1.08e10 - 1.08e10i)T - 1.12e20iT^{2} \)
17 \( 1 + (-4.48e10 - 4.48e10i)T + 1.40e22iT^{2} \)
19 \( 1 + 3.49e11iT - 1.04e23T^{2} \)
23 \( 1 + (-9.76e11 + 9.76e11i)T - 3.24e24iT^{2} \)
29 \( 1 + 2.06e13iT - 2.10e26T^{2} \)
31 \( 1 - 1.69e13T + 6.99e26T^{2} \)
37 \( 1 + (9.71e13 + 9.71e13i)T + 1.68e28iT^{2} \)
41 \( 1 - 2.27e14T + 1.07e29T^{2} \)
43 \( 1 + (-5.68e14 + 5.68e14i)T - 2.52e29iT^{2} \)
47 \( 1 + (-1.52e14 - 1.52e14i)T + 1.25e30iT^{2} \)
53 \( 1 + (1.38e15 - 1.38e15i)T - 1.08e31iT^{2} \)
59 \( 1 - 9.12e15iT - 7.50e31T^{2} \)
61 \( 1 + 9.20e15T + 1.36e32T^{2} \)
67 \( 1 + (-2.50e16 - 2.50e16i)T + 7.40e32iT^{2} \)
71 \( 1 - 7.31e15T + 2.10e33T^{2} \)
73 \( 1 + (-1.46e16 + 1.46e16i)T - 3.46e33iT^{2} \)
79 \( 1 - 1.58e17iT - 1.43e34T^{2} \)
83 \( 1 + (-1.61e17 + 1.61e17i)T - 3.49e34iT^{2} \)
89 \( 1 + 2.29e17iT - 1.22e35T^{2} \)
97 \( 1 + (-1.30e17 - 1.30e17i)T + 5.77e35iT^{2} \)
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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

−15.47117396430951385971223086314, −14.18870930096609929931985205702, −13.09990048453661645739808792946, −12.11800827256218814217576704062, −9.047748620311745585500490344506, −7.66408975026996835606947549092, −6.85613304245188063366321471736, −4.07698888876322726180031619577, −2.59055197177251470073576155211, −0.54145104970842437376329615526, 2.76535467909683955626955859506, 3.41897381617851833997536920240, 5.00625333875180864627312012222, 7.903797661440814543397003795267, 9.514419311890539150146933290998, 10.62190564694141800145402745237, 12.51666096841470098033456782445, 14.32837629755935066739764750527, 15.21825381338190031475688728345, 16.05578731729545677488834331707

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