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Properties

Label 2-10-5.3-c8-0-2
Degree $2$
Conductor $10$
Sign $0.699 + 0.715i$
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 + (−25.9 − 25.9i)3-s − 128i·4-s + (535. − 322. i)5-s + 415.·6-s + (2.03e3 − 2.03e3i)7-s + (1.02e3 + 1.02e3i)8-s − 5.21e3i·9-s + (−1.70e3 + 6.86e3i)10-s − 1.52e4·11-s + (−3.32e3 + 3.32e3i)12-s + (8.87e3 + 8.87e3i)13-s + 3.24e4i·14-s + (−2.22e4 − 5.52e3i)15-s − 1.63e4·16-s + (4.60e4 − 4.60e4i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.320 − 0.320i)3-s − 0.5i·4-s + (0.856 − 0.516i)5-s + 0.320·6-s + (0.845 − 0.845i)7-s + (0.250 + 0.250i)8-s − 0.794i·9-s + (−0.170 + 0.686i)10-s − 1.04·11-s + (−0.160 + 0.160i)12-s + (0.310 + 0.310i)13-s + 0.845i·14-s + (−0.439 − 0.109i)15-s − 0.250·16-s + (0.551 − 0.551i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.715i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.699 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.699 + 0.715i$
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: \((2,\ 10,\ (\ :4),\ 0.699 + 0.715i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.08087 - 0.454895i\)
\(L(\frac12)\) \(\approx\) \(1.08087 - 0.454895i\)
\(L(5)\) 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 \( 1 + (-535. + 322. i)T \)
good3 \( 1 + (25.9 + 25.9i)T + 6.56e3iT^{2} \)
7 \( 1 + (-2.03e3 + 2.03e3i)T - 5.76e6iT^{2} \)
11 \( 1 + 1.52e4T + 2.14e8T^{2} \)
13 \( 1 + (-8.87e3 - 8.87e3i)T + 8.15e8iT^{2} \)
17 \( 1 + (-4.60e4 + 4.60e4i)T - 6.97e9iT^{2} \)
19 \( 1 + 7.32e4iT - 1.69e10T^{2} \)
23 \( 1 + (-2.63e5 - 2.63e5i)T + 7.83e10iT^{2} \)
29 \( 1 - 1.20e6iT - 5.00e11T^{2} \)
31 \( 1 + 1.40e6T + 8.52e11T^{2} \)
37 \( 1 + (-1.51e6 + 1.51e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 2.03e6T + 7.98e12T^{2} \)
43 \( 1 + (4.41e5 + 4.41e5i)T + 1.16e13iT^{2} \)
47 \( 1 + (3.91e6 - 3.91e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (-8.72e6 - 8.72e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 4.77e6iT - 1.46e14T^{2} \)
61 \( 1 - 2.24e7T + 1.91e14T^{2} \)
67 \( 1 + (1.55e7 - 1.55e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 3.03e7T + 6.45e14T^{2} \)
73 \( 1 + (1.18e7 + 1.18e7i)T + 8.06e14iT^{2} \)
79 \( 1 + 2.53e7iT - 1.51e15T^{2} \)
83 \( 1 + (-1.13e7 - 1.13e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 1.02e7iT - 3.93e15T^{2} \)
97 \( 1 + (-4.11e7 + 4.11e7i)T - 7.83e15iT^{2} \)
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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

−18.26956493250383704184179346780, −17.57676066475171979043589777511, −16.34558823557599653074846658234, −14.48880283528666747278736147216, −13.05163793567470218738733518796, −10.94366761240132015743559132895, −9.209487462885946197681305327800, −7.29154380092448956159123600654, −5.34262322075842697755823825540, −1.09122246047220759063044200525, 2.25191905571300630546010570399, 5.45269018885786286140952283594, 8.154952320465890285594692637568, 10.12109764209585831458647257869, 11.19450287195086172685431149576, 13.11309136458685725931638834512, 14.91159500540444449651290469259, 16.66697653822403080057316495135, 18.02987507352447478999831013558, 18.85516239329005465625546631500

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