Properties

Label 2-490-1.1-c1-0-7
Degree $2$
Conductor $490$
Sign $1$
Analytic cond. $3.91266$
Root an. cond. $1.97804$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3.41·3-s + 4-s + 5-s − 3.41·6-s − 8-s + 8.65·9-s − 10-s − 0.828·11-s + 3.41·12-s − 4.82·13-s + 3.41·15-s + 16-s + 2.58·17-s − 8.65·18-s + 0.585·19-s + 20-s + 0.828·22-s − 1.17·23-s − 3.41·24-s + 25-s + 4.82·26-s + 19.3·27-s − 4.82·29-s − 3.41·30-s − 2.82·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.97·3-s + 0.5·4-s + 0.447·5-s − 1.39·6-s − 0.353·8-s + 2.88·9-s − 0.316·10-s − 0.249·11-s + 0.985·12-s − 1.33·13-s + 0.881·15-s + 0.250·16-s + 0.627·17-s − 2.04·18-s + 0.134·19-s + 0.223·20-s + 0.176·22-s − 0.244·23-s − 0.696·24-s + 0.200·25-s + 0.946·26-s + 3.71·27-s − 0.896·29-s − 0.623·30-s − 0.508·31-s − 0.176·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.071613186\)
\(L(\frac12)\) \(\approx\) \(2.071613186\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
good3 \( 1 - 3.41T + 3T^{2} \)
11 \( 1 + 0.828T + 11T^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
17 \( 1 - 2.58T + 17T^{2} \)
19 \( 1 - 0.585T + 19T^{2} \)
23 \( 1 + 1.17T + 23T^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 + 3.07T + 41T^{2} \)
43 \( 1 + 8.82T + 43T^{2} \)
47 \( 1 - 5.17T + 47T^{2} \)
53 \( 1 - 6.48T + 53T^{2} \)
59 \( 1 - 8.58T + 59T^{2} \)
61 \( 1 - 9.31T + 61T^{2} \)
67 \( 1 - 1.65T + 67T^{2} \)
71 \( 1 + 4.48T + 71T^{2} \)
73 \( 1 + 9.41T + 73T^{2} \)
79 \( 1 + 6.82T + 79T^{2} \)
83 \( 1 + 2.24T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 7.75T + 97T^{2} \)
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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

−10.28971780065301490311421084737, −9.943249755000821756197060500778, −9.117572308403497337302090990962, −8.410575894350371954381270306616, −7.51000711774831488048519746298, −6.97136286400532135914052356201, −5.21403970872239700718804396046, −3.73518003555372199495859711444, −2.66773972862876738845706573219, −1.77576856829648359670824497850, 1.77576856829648359670824497850, 2.66773972862876738845706573219, 3.73518003555372199495859711444, 5.21403970872239700718804396046, 6.97136286400532135914052356201, 7.51000711774831488048519746298, 8.410575894350371954381270306616, 9.117572308403497337302090990962, 9.943249755000821756197060500778, 10.28971780065301490311421084737

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