Properties

Label 2-10-5.4-c11-0-5
Degree $2$
Conductor $10$
Sign $-0.973 - 0.228i$
Analytic cond. $7.68343$
Root an. cond. $2.77190$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 32i·2-s − 532. i·3-s − 1.02e3·4-s + (1.59e3 − 6.80e3i)5-s − 1.70e4·6-s + 2.44e4i·7-s + 3.27e4i·8-s − 1.06e5·9-s + (−2.17e5 − 5.10e4i)10-s − 6.69e5·11-s + 5.44e5i·12-s + 5.79e5i·13-s + 7.83e5·14-s + (−3.62e6 − 8.48e5i)15-s + 1.04e6·16-s − 7.85e6i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.26i·3-s − 0.5·4-s + (0.228 − 0.973i)5-s − 0.894·6-s + 0.550i·7-s + 0.353i·8-s − 0.598·9-s + (−0.688 − 0.161i)10-s − 1.25·11-s + 0.632i·12-s + 0.432i·13-s + 0.389·14-s + (−1.23 − 0.288i)15-s + 0.250·16-s − 1.34i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.973 - 0.228i$
Analytic conductor: \(7.68343\)
Root analytic conductor: \(2.77190\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :11/2),\ -0.973 - 0.228i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.147292 + 1.27385i\)
\(L(\frac12)\) \(\approx\) \(0.147292 + 1.27385i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 32iT \)
5 \( 1 + (-1.59e3 + 6.80e3i)T \)
good3 \( 1 + 532. iT - 1.77e5T^{2} \)
7 \( 1 - 2.44e4iT - 1.97e9T^{2} \)
11 \( 1 + 6.69e5T + 2.85e11T^{2} \)
13 \( 1 - 5.79e5iT - 1.79e12T^{2} \)
17 \( 1 + 7.85e6iT - 3.42e13T^{2} \)
19 \( 1 - 1.81e7T + 1.16e14T^{2} \)
23 \( 1 + 1.97e7iT - 9.52e14T^{2} \)
29 \( 1 + 2.02e8T + 1.22e16T^{2} \)
31 \( 1 - 1.02e8T + 2.54e16T^{2} \)
37 \( 1 + 5.02e8iT - 1.77e17T^{2} \)
41 \( 1 - 3.44e8T + 5.50e17T^{2} \)
43 \( 1 - 1.43e9iT - 9.29e17T^{2} \)
47 \( 1 + 1.38e9iT - 2.47e18T^{2} \)
53 \( 1 + 2.89e8iT - 9.26e18T^{2} \)
59 \( 1 - 2.47e9T + 3.01e19T^{2} \)
61 \( 1 - 9.16e7T + 4.35e19T^{2} \)
67 \( 1 - 3.09e7iT - 1.22e20T^{2} \)
71 \( 1 - 2.42e10T + 2.31e20T^{2} \)
73 \( 1 + 3.55e9iT - 3.13e20T^{2} \)
79 \( 1 - 1.16e10T + 7.47e20T^{2} \)
83 \( 1 - 2.33e10iT - 1.28e21T^{2} \)
89 \( 1 - 1.02e11T + 2.77e21T^{2} \)
97 \( 1 + 5.87e10iT - 7.15e21T^{2} \)
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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

−18.00910224863678527329876879570, −16.14552477139428976875558225053, −13.73584877030380804469992736400, −12.80171197392832559384323944194, −11.71561964779402905227548977969, −9.396906711323708128250127533502, −7.70966936221151784144915269511, −5.31332013620488253374712914398, −2.29550708362711598235268447601, −0.68642359667407362647511778914, 3.55183640742035152665427837019, 5.47718555325092305754895097621, 7.59135748171717876928684857289, 9.803585997656866850439421241991, 10.75812317857135117298690778561, 13.50603399233296482936873549656, 14.99368393438504922038159646635, 15.79110148913278842434673570483, 17.25121688714027371339673642786, 18.61428196793718808981146953485

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