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Properties

Label 2-10-5.3-c8-0-0
Degree $2$
Conductor $10$
Sign $-0.783 - 0.621i$
Analytic cond. $4.07378$
Root an. cond. $2.01836$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8 + 8i)2-s + (52.9 + 52.9i)3-s − 128i·4-s + (−490. + 387. i)5-s − 847.·6-s + (−2.62e3 + 2.62e3i)7-s + (1.02e3 + 1.02e3i)8-s − 953. i·9-s + (822. − 7.02e3i)10-s + 5.37e3·11-s + (6.77e3 − 6.77e3i)12-s + (2.79e4 + 2.79e4i)13-s − 4.19e4i·14-s + (−4.64e4 − 5.44e3i)15-s − 1.63e4·16-s + (5.34e4 − 5.34e4i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.653 + 0.653i)3-s − 0.5i·4-s + (−0.784 + 0.620i)5-s − 0.653·6-s + (−1.09 + 1.09i)7-s + (0.250 + 0.250i)8-s − 0.145i·9-s + (0.0822 − 0.702i)10-s + 0.366·11-s + (0.326 − 0.326i)12-s + (0.979 + 0.979i)13-s − 1.09i·14-s + (−0.918 − 0.107i)15-s − 0.250·16-s + (0.639 − 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.783 - 0.621i$
Analytic conductor: \(4.07378\)
Root analytic conductor: \(2.01836\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :4),\ -0.783 - 0.621i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.334505 + 0.960670i\)
\(L(\frac12)\) \(\approx\) \(0.334505 + 0.960670i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (8 - 8i)T \)
5 \( 1 + (490. - 387. i)T \)
good3 \( 1 + (-52.9 - 52.9i)T + 6.56e3iT^{2} \)
7 \( 1 + (2.62e3 - 2.62e3i)T - 5.76e6iT^{2} \)
11 \( 1 - 5.37e3T + 2.14e8T^{2} \)
13 \( 1 + (-2.79e4 - 2.79e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (-5.34e4 + 5.34e4i)T - 6.97e9iT^{2} \)
19 \( 1 - 1.44e5iT - 1.69e10T^{2} \)
23 \( 1 + (-5.21e4 - 5.21e4i)T + 7.83e10iT^{2} \)
29 \( 1 - 4.17e4iT - 5.00e11T^{2} \)
31 \( 1 - 2.44e5T + 8.52e11T^{2} \)
37 \( 1 + (2.00e6 - 2.00e6i)T - 3.51e12iT^{2} \)
41 \( 1 + 4.17e6T + 7.98e12T^{2} \)
43 \( 1 + (-4.01e6 - 4.01e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (-2.26e6 + 2.26e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (3.72e6 + 3.72e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 1.12e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.02e7T + 1.91e14T^{2} \)
67 \( 1 + (1.30e7 - 1.30e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 2.61e7T + 6.45e14T^{2} \)
73 \( 1 + (1.85e7 + 1.85e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 3.07e6iT - 1.51e15T^{2} \)
83 \( 1 + (-2.41e7 - 2.41e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 5.76e6iT - 3.93e15T^{2} \)
97 \( 1 + (-2.86e7 + 2.86e7i)T - 7.83e15iT^{2} \)
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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

−19.22039782453345498122489228997, −18.55225175777111055126870650811, −16.25195716448355735395762076240, −15.49118348210720808768659522907, −14.30501672796265016763968391433, −11.88678472185384620331631544692, −9.815915125807387521427744484699, −8.610619082343812331927088025774, −6.47798515963746562097895777788, −3.43396587058692779369520740118, 0.801098543784933523304306953121, 3.52661646363352644481874717553, 7.31654700935727973072703631691, 8.687319578953972657443850994476, 10.62947254230871342329311705112, 12.62228437719150411911918321604, 13.54302968260372092947213546653, 15.85370592360411122446267774804, 17.13582691403347046369645163075, 19.01291095096740593356407572430

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