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Properties

Label 2-490-245.239-c1-0-0
Degree $2$
Conductor $490$
Sign $-0.545 + 0.838i$
Analytic cond. $3.91266$
Root an. cond. $1.97804$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.781 − 0.623i)2-s + (−1.23 + 2.56i)3-s + (0.222 + 0.974i)4-s + (−1.72 + 1.42i)5-s + (2.56 − 1.23i)6-s + (−1.42 + 2.22i)7-s + (0.433 − 0.900i)8-s + (−3.17 − 3.97i)9-s + (2.23 − 0.0361i)10-s + (−3.14 + 3.94i)11-s + (−2.77 − 0.632i)12-s + (−3.59 − 2.86i)13-s + (2.50 − 0.854i)14-s + (−1.51 − 6.17i)15-s + (−0.900 + 0.433i)16-s + (6.93 + 1.58i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (−0.712 + 1.47i)3-s + (0.111 + 0.487i)4-s + (−0.771 + 0.636i)5-s + (1.04 − 0.503i)6-s + (−0.538 + 0.842i)7-s + (0.153 − 0.318i)8-s + (−1.05 − 1.32i)9-s + (0.707 − 0.0114i)10-s + (−0.948 + 1.18i)11-s + (−0.800 − 0.182i)12-s + (−0.996 − 0.794i)13-s + (0.669 − 0.228i)14-s + (−0.391 − 1.59i)15-s + (−0.225 + 0.108i)16-s + (1.68 + 0.384i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.545 + 0.838i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.545 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.108856 - 0.200651i\)
\(L(\frac12)\) \(\approx\) \(0.108856 - 0.200651i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.781 + 0.623i)T \)
5 \( 1 + (1.72 - 1.42i)T \)
7 \( 1 + (1.42 - 2.22i)T \)
good3 \( 1 + (1.23 - 2.56i)T + (-1.87 - 2.34i)T^{2} \)
11 \( 1 + (3.14 - 3.94i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (3.59 + 2.86i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-6.93 - 1.58i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 - 6.75T + 19T^{2} \)
23 \( 1 + (0.428 - 0.0977i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.196 - 0.863i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 0.641T + 31T^{2} \)
37 \( 1 + (10.1 + 2.32i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + (-4.74 - 2.28i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (0.343 + 0.713i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (0.250 + 0.199i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (3.92 - 0.896i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (4.16 - 2.00i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-0.950 + 4.16i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 0.809iT - 67T^{2} \)
71 \( 1 + (-1.44 - 6.30i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-2.10 + 1.67i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + (5.09 - 4.05i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (-6.77 - 8.49i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 0.945iT - 97T^{2} \)
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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

−11.38613846688400955002197059460, −10.28442008861221492941146172420, −10.09342751317297195192190509437, −9.355829127320043299182980754596, −7.966403977992416038891307892921, −7.23203951725811096301929051993, −5.62347023265887580216918850232, −4.96613122144179795703222619990, −3.56868844447011261490189934023, −2.82540293762580162716066476003, 0.20430935350139656271172851838, 1.19931154791899837560763031669, 3.21007534230905470000214616889, 5.07144813232073976010785400922, 5.80932869312625157916683328594, 7.06939826988157605249587416166, 7.51829925642652418300078248595, 8.110687698901937301271186626278, 9.390694160041536229665750613301, 10.41861126465490761149556107184

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