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Properties

Label 2-490-5.4-c3-0-31
Degree $2$
Conductor $490$
Sign $-0.117 + 0.993i$
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 − 3.17i·3-s − 4·4-s + (−11.1 − 1.30i)5-s − 6.34·6-s + 8i·8-s + 16.9·9-s + (−2.61 + 22.2i)10-s + 50.2·11-s + 12.6i·12-s − 17.8i·13-s + (−4.15 + 35.2i)15-s + 16·16-s + 93.9i·17-s − 33.8i·18-s + 84.9·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.611i·3-s − 0.5·4-s + (−0.993 − 0.117i)5-s − 0.432·6-s + 0.353i·8-s + 0.626·9-s + (−0.0828 + 0.702i)10-s + 1.37·11-s + 0.305i·12-s − 0.381i·13-s + (−0.0715 + 0.606i)15-s + 0.250·16-s + 1.33i·17-s − 0.443i·18-s + 1.02·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.117 + 0.993i$
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: \((2,\ 490,\ (\ :3/2),\ -0.117 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.737297015\)
\(L(\frac12)\) \(\approx\) \(1.737297015\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.1 + 1.30i)T \)
7 \( 1 \)
good3 \( 1 + 3.17iT - 27T^{2} \)
11 \( 1 - 50.2T + 1.33e3T^{2} \)
13 \( 1 + 17.8iT - 2.19e3T^{2} \)
17 \( 1 - 93.9iT - 4.91e3T^{2} \)
19 \( 1 - 84.9T + 6.85e3T^{2} \)
23 \( 1 - 119. iT - 1.21e4T^{2} \)
29 \( 1 + 165.T + 2.43e4T^{2} \)
31 \( 1 - 134.T + 2.97e4T^{2} \)
37 \( 1 + 129. iT - 5.06e4T^{2} \)
41 \( 1 - 118.T + 6.89e4T^{2} \)
43 \( 1 + 533. iT - 7.95e4T^{2} \)
47 \( 1 + 142. iT - 1.03e5T^{2} \)
53 \( 1 + 549. iT - 1.48e5T^{2} \)
59 \( 1 - 566.T + 2.05e5T^{2} \)
61 \( 1 + 523.T + 2.26e5T^{2} \)
67 \( 1 - 503. iT - 3.00e5T^{2} \)
71 \( 1 - 179.T + 3.57e5T^{2} \)
73 \( 1 - 1.10e3iT - 3.89e5T^{2} \)
79 \( 1 - 499.T + 4.93e5T^{2} \)
83 \( 1 + 626. iT - 5.71e5T^{2} \)
89 \( 1 - 721.T + 7.04e5T^{2} \)
97 \( 1 + 543. iT - 9.12e5T^{2} \)
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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

−10.41120795650577968889388612248, −9.464355134565160796286718150667, −8.518460502874663864492356257813, −7.61588026735400977605631328667, −6.82859001502382894041779810154, −5.51957886587438345313244831314, −4.07578888412034203119920276149, −3.56550284572364395342375359879, −1.78285035016366779133305473855, −0.799824896676620707206783095729, 0.958869273931895434883901056306, 3.22007762798268091748441181328, 4.27267821087105557917848362375, 4.82667982721168831499800482932, 6.36502144790218130809136969627, 7.14695725902614094228223675530, 7.921493588434647263863619710101, 9.200355974652036682126152951895, 9.514111774222514269983877916326, 10.79483133046629681322545546961

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