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Properties

Label 2-10-5.2-c18-0-0
Degree $2$
Conductor $10$
Sign $0.838 - 0.544i$
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 + (−2.24e4 + 2.24e4i)3-s + 1.31e5i·4-s + (−1.41e6 − 1.35e6i)5-s + 1.15e7·6-s + (−5.59e7 − 5.59e7i)7-s + (3.35e7 − 3.35e7i)8-s − 6.23e8i·9-s + (1.57e7 + 7.06e8i)10-s − 1.86e9·11-s + (−2.94e9 − 2.94e9i)12-s + (−7.54e9 + 7.54e9i)13-s + 2.86e10i·14-s + (6.20e10 − 1.37e9i)15-s − 1.71e10·16-s + (−6.21e10 − 6.21e10i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−1.14 + 1.14i)3-s + 0.5i·4-s + (−0.722 − 0.691i)5-s + 1.14·6-s + (−1.38 − 1.38i)7-s + (0.250 − 0.250i)8-s − 1.60i·9-s + (0.0157 + 0.706i)10-s − 0.791·11-s + (−0.571 − 0.571i)12-s + (−0.711 + 0.711i)13-s + 1.38i·14-s + (1.61 − 0.0358i)15-s − 0.250·16-s + (−0.524 − 0.524i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.838 - 0.544i$
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: \((2,\ 10,\ (\ :9),\ 0.838 - 0.544i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.0965236 + 0.0285825i\)
\(L(\frac12)\) \(\approx\) \(0.0965236 + 0.0285825i\)
\(L(10)\) 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 \( 1 + (1.41e6 + 1.35e6i)T \)
good3 \( 1 + (2.24e4 - 2.24e4i)T - 3.87e8iT^{2} \)
7 \( 1 + (5.59e7 + 5.59e7i)T + 1.62e15iT^{2} \)
11 \( 1 + 1.86e9T + 5.55e18T^{2} \)
13 \( 1 + (7.54e9 - 7.54e9i)T - 1.12e20iT^{2} \)
17 \( 1 + (6.21e10 + 6.21e10i)T + 1.40e22iT^{2} \)
19 \( 1 + 2.58e11iT - 1.04e23T^{2} \)
23 \( 1 + (1.18e12 - 1.18e12i)T - 3.24e24iT^{2} \)
29 \( 1 - 1.23e13iT - 2.10e26T^{2} \)
31 \( 1 - 1.56e12T + 6.99e26T^{2} \)
37 \( 1 + (6.47e13 + 6.47e13i)T + 1.68e28iT^{2} \)
41 \( 1 - 3.10e13T + 1.07e29T^{2} \)
43 \( 1 + (4.19e13 - 4.19e13i)T - 2.52e29iT^{2} \)
47 \( 1 + (1.22e15 + 1.22e15i)T + 1.25e30iT^{2} \)
53 \( 1 + (-1.53e15 + 1.53e15i)T - 1.08e31iT^{2} \)
59 \( 1 + 7.41e15iT - 7.50e31T^{2} \)
61 \( 1 - 1.51e16T + 1.36e32T^{2} \)
67 \( 1 + (8.12e15 + 8.12e15i)T + 7.40e32iT^{2} \)
71 \( 1 + 2.90e16T + 2.10e33T^{2} \)
73 \( 1 + (3.39e16 - 3.39e16i)T - 3.46e33iT^{2} \)
79 \( 1 + 1.44e17iT - 1.43e34T^{2} \)
83 \( 1 + (-1.04e17 + 1.04e17i)T - 3.49e34iT^{2} \)
89 \( 1 + 2.22e17iT - 1.22e35T^{2} \)
97 \( 1 + (-2.70e17 - 2.70e17i)T + 5.77e35iT^{2} \)
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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

−16.38005599268216755955892397753, −15.91366827089311670044652260399, −13.10369117595169958340264898380, −11.64012527767500357094696851248, −10.42232849680696063429452510786, −9.418299154638190558470321502738, −7.06858456722552993949137919012, −4.79357238090980333396214504867, −3.63488448717687644603495707794, −0.39396320694904692381469557951, 0.13698368957021480781603595075, 2.50290198515904121216795507988, 5.72660968703799243254860057917, 6.62557539362688545659952038950, 8.024640939996084192776804981644, 10.26023100858878167516410473539, 11.92643797195828672852136996683, 12.86186561334211709756657141855, 15.21631728358013531617413287012, 16.26987147155314634987217901565

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