Properties

Label 2-490-5.4-c3-0-29
Degree $2$
Conductor $490$
Sign $0.447 + 0.894i$
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 + 7i·3-s − 4·4-s + (−10 + 5i)5-s + 14·6-s + 8i·8-s − 22·9-s + (10 + 20i)10-s − 37·11-s − 28i·12-s − 51i·13-s + (−35 − 70i)15-s + 16·16-s + 41i·17-s + 44i·18-s − 108·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.34i·3-s − 0.5·4-s + (−0.894 + 0.447i)5-s + 0.952·6-s + 0.353i·8-s − 0.814·9-s + (0.316 + 0.632i)10-s − 1.01·11-s − 0.673i·12-s − 1.08i·13-s + (−0.602 − 1.20i)15-s + 0.250·16-s + 0.584i·17-s + 0.576i·18-s − 1.30·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.447 + 0.894i$
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: \((2,\ 490,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7585708601\)
\(L(\frac12)\) \(\approx\) \(0.7585708601\)
\(L(\frac{5}{2})\) 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 \( 1 + (10 - 5i)T \)
7 \( 1 \)
good3 \( 1 - 7iT - 27T^{2} \)
11 \( 1 + 37T + 1.33e3T^{2} \)
13 \( 1 + 51iT - 2.19e3T^{2} \)
17 \( 1 - 41iT - 4.91e3T^{2} \)
19 \( 1 + 108T + 6.85e3T^{2} \)
23 \( 1 + 70iT - 1.21e4T^{2} \)
29 \( 1 - 249T + 2.43e4T^{2} \)
31 \( 1 - 134T + 2.97e4T^{2} \)
37 \( 1 - 334iT - 5.06e4T^{2} \)
41 \( 1 + 206T + 6.89e4T^{2} \)
43 \( 1 + 376iT - 7.95e4T^{2} \)
47 \( 1 + 287iT - 1.03e5T^{2} \)
53 \( 1 + 6iT - 1.48e5T^{2} \)
59 \( 1 + 2T + 2.05e5T^{2} \)
61 \( 1 - 940T + 2.26e5T^{2} \)
67 \( 1 + 106iT - 3.00e5T^{2} \)
71 \( 1 - 456T + 3.57e5T^{2} \)
73 \( 1 + 650iT - 3.89e5T^{2} \)
79 \( 1 - 1.23e3T + 4.93e5T^{2} \)
83 \( 1 + 428iT - 5.71e5T^{2} \)
89 \( 1 + 220T + 7.04e5T^{2} \)
97 \( 1 + 1.05e3iT - 9.12e5T^{2} \)
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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

−10.39147450357880000793897853948, −10.06534631538059304254435294140, −8.455793591411756842923583508376, −8.242133549592249647955203628102, −6.64487809452868843139584167533, −5.19701279982007051140619635344, −4.43398132836421108305922104039, −3.50332081764238534199157002609, −2.62107990296773824875865032064, −0.30722995468467903149198132616, 0.936028658556281577130460374182, 2.45957747222641861007880517789, 4.14376528456115719513893016371, 5.12336014758603854763038433786, 6.44901021091649594988061890626, 7.03965472727530106930365939734, 7.981984847407068270269091721237, 8.390646226696373737197570646662, 9.533275396470327408956705544434, 10.88984858452741850413261644410

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