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Properties

Label 2-490-5.4-c3-0-41
Degree $2$
Conductor $490$
Sign $-0.263 + 0.964i$
Analytic cond. $28.9109$
Root an. cond. $5.37688$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 6.79i·3-s − 4·4-s + (10.7 + 2.94i)5-s − 13.5·6-s + 8i·8-s − 19.1·9-s + (5.88 − 21.5i)10-s + 69.2·11-s + 27.1i·12-s + 28.3i·13-s + (19.9 − 73.2i)15-s + 16·16-s + 11.3i·17-s + 38.2i·18-s + 74.2·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.30i·3-s − 0.5·4-s + (0.964 + 0.263i)5-s − 0.924·6-s + 0.353i·8-s − 0.708·9-s + (0.186 − 0.682i)10-s + 1.89·11-s + 0.653i·12-s + 0.605i·13-s + (0.344 − 1.26i)15-s + 0.250·16-s + 0.161i·17-s + 0.500i·18-s + 0.896·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.263 + 0.964i$
Analytic conductor: \(28.9109\)
Root analytic conductor: \(5.37688\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :3/2),\ -0.263 + 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.642825593\)
\(L(\frac12)\) \(\approx\) \(2.642825593\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
5 \( 1 + (-10.7 - 2.94i)T \)
7 \( 1 \)
good3 \( 1 + 6.79iT - 27T^{2} \)
11 \( 1 - 69.2T + 1.33e3T^{2} \)
13 \( 1 - 28.3iT - 2.19e3T^{2} \)
17 \( 1 - 11.3iT - 4.91e3T^{2} \)
19 \( 1 - 74.2T + 6.85e3T^{2} \)
23 \( 1 - 66.6iT - 1.21e4T^{2} \)
29 \( 1 - 134.T + 2.43e4T^{2} \)
31 \( 1 - 160.T + 2.97e4T^{2} \)
37 \( 1 + 284. iT - 5.06e4T^{2} \)
41 \( 1 + 149.T + 6.89e4T^{2} \)
43 \( 1 - 513. iT - 7.95e4T^{2} \)
47 \( 1 + 411. iT - 1.03e5T^{2} \)
53 \( 1 - 354. iT - 1.48e5T^{2} \)
59 \( 1 - 210.T + 2.05e5T^{2} \)
61 \( 1 + 587.T + 2.26e5T^{2} \)
67 \( 1 - 1.03e3iT - 3.00e5T^{2} \)
71 \( 1 + 1.03e3T + 3.57e5T^{2} \)
73 \( 1 + 743. iT - 3.89e5T^{2} \)
79 \( 1 + 609.T + 4.93e5T^{2} \)
83 \( 1 + 1.14e3iT - 5.71e5T^{2} \)
89 \( 1 + 735.T + 7.04e5T^{2} \)
97 \( 1 - 333. iT - 9.12e5T^{2} \)
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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

−10.24519528655758526813883186810, −9.413921790697875181921061351893, −8.709622359292009416510600580772, −7.37711406525809190334493624533, −6.58640387422755987783280599713, −5.86508907833017332374562493874, −4.34252883424502996446143046391, −2.96195812310249770788243203469, −1.69388957067861105106622343801, −1.14982087010032639628013698355, 1.17004862600813045799085994247, 3.17703585027987588029305306986, 4.33028805601388560835772234837, 5.08128499806278706966148780586, 6.09728888730255158146144190745, 6.90035750414970712273692492230, 8.436349732349409652119670988991, 9.123220312949570146161387356935, 9.817473519994742442334913620088, 10.36417014214742457385719731564

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