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Properties

Label 2-10-5.4-c13-0-3
Degree $2$
Conductor $10$
Sign $0.999 + 0.00780i$
Analytic cond. $10.7230$
Root an. cond. $3.27461$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 64i·2-s − 948. i·3-s − 4.09e3·4-s + (−272. + 3.49e4i)5-s + 6.07e4·6-s − 4.54e5i·7-s − 2.62e5i·8-s + 6.93e5·9-s + (−2.23e6 − 1.74e4i)10-s + 1.09e7·11-s + 3.88e6i·12-s + 1.27e7i·13-s + 2.90e7·14-s + (3.31e7 + 2.58e5i)15-s + 1.67e7·16-s − 1.16e8i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.751i·3-s − 0.5·4-s + (−0.00780 + 0.999i)5-s + 0.531·6-s − 1.46i·7-s − 0.353i·8-s + 0.435·9-s + (−0.707 − 0.00551i)10-s + 1.85·11-s + 0.375i·12-s + 0.732i·13-s + 1.03·14-s + (0.751 + 0.00586i)15-s + 0.250·16-s − 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00780i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.999 + 0.00780i$
Analytic conductor: \(10.7230\)
Root analytic conductor: \(3.27461\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :13/2),\ 0.999 + 0.00780i)\)

Particular Values

\(L(7)\) \(\approx\) \(1.84961 - 0.00721548i\)
\(L(\frac12)\) \(\approx\) \(1.84961 - 0.00721548i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 64iT \)
5 \( 1 + (272. - 3.49e4i)T \)
good3 \( 1 + 948. iT - 1.59e6T^{2} \)
7 \( 1 + 4.54e5iT - 9.68e10T^{2} \)
11 \( 1 - 1.09e7T + 3.45e13T^{2} \)
13 \( 1 - 1.27e7iT - 3.02e14T^{2} \)
17 \( 1 + 1.16e8iT - 9.90e15T^{2} \)
19 \( 1 - 2.73e8T + 4.20e16T^{2} \)
23 \( 1 + 1.52e8iT - 5.04e17T^{2} \)
29 \( 1 - 8.89e8T + 1.02e19T^{2} \)
31 \( 1 + 4.78e9T + 2.44e19T^{2} \)
37 \( 1 - 5.25e9iT - 2.43e20T^{2} \)
41 \( 1 + 2.70e9T + 9.25e20T^{2} \)
43 \( 1 - 8.03e9iT - 1.71e21T^{2} \)
47 \( 1 + 3.35e10iT - 5.46e21T^{2} \)
53 \( 1 - 1.38e11iT - 2.60e22T^{2} \)
59 \( 1 - 3.13e11T + 1.04e23T^{2} \)
61 \( 1 - 7.15e11T + 1.61e23T^{2} \)
67 \( 1 + 4.70e11iT - 5.48e23T^{2} \)
71 \( 1 + 4.88e11T + 1.16e24T^{2} \)
73 \( 1 - 1.10e12iT - 1.67e24T^{2} \)
79 \( 1 + 2.70e12T + 4.66e24T^{2} \)
83 \( 1 - 1.70e12iT - 8.87e24T^{2} \)
89 \( 1 - 7.89e11T + 2.19e25T^{2} \)
97 \( 1 + 4.98e12iT - 6.73e25T^{2} \)
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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

−17.59568383547192357212507528667, −16.32464676242470945779293430485, −14.34685877870222412433274886880, −13.71454967597905991396891076687, −11.62547105354115923561169680117, −9.705821545361806323504124817608, −7.25179658944734329366348982812, −6.77690173979815862989230014248, −3.94118744499985420328745118990, −1.09417174697461626629017050982, 1.44317874118284248237121164903, 3.81833346974654758387430895187, 5.46068772788961261188598811320, 8.754051409991038073735252563781, 9.675066813121861567467340777191, 11.70634455245492478062465578249, 12.74285366086412861441871278203, 14.80904812481888424020741031207, 16.13277563963032902385575781685, 17.64027569812877784985258396501

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