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Properties

Label 2-490-245.103-c1-0-0
Degree $2$
Conductor $490$
Sign $-0.791 - 0.610i$
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  + (0.593 − 0.804i)2-s + (−3.24 + 0.614i)3-s + (−0.294 − 0.955i)4-s + (2.20 + 0.375i)5-s + (−1.43 + 2.97i)6-s + (−2.41 − 1.08i)7-s + (−0.943 − 0.330i)8-s + (7.36 − 2.89i)9-s + (1.61 − 1.55i)10-s + (−0.382 + 0.975i)11-s + (1.54 + 2.92i)12-s + (−2.24 − 0.253i)13-s + (−2.30 + 1.29i)14-s + (−7.38 + 0.136i)15-s + (−0.826 + 0.563i)16-s + (−4.74 − 0.177i)17-s + ⋯
L(s)  = 1  + (0.419 − 0.568i)2-s + (−1.87 + 0.354i)3-s + (−0.147 − 0.477i)4-s + (0.985 + 0.167i)5-s + (−0.585 + 1.21i)6-s + (−0.911 − 0.411i)7-s + (−0.333 − 0.116i)8-s + (2.45 − 0.963i)9-s + (0.509 − 0.490i)10-s + (−0.115 + 0.293i)11-s + (0.445 + 0.843i)12-s + (−0.623 − 0.0702i)13-s + (−0.616 + 0.345i)14-s + (−1.90 + 0.0351i)15-s + (−0.206 + 0.140i)16-s + (−1.15 − 0.0430i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.791 - 0.610i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.791 - 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00834446 + 0.0244893i\)
\(L(\frac12)\) \(\approx\) \(0.00834446 + 0.0244893i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.593 + 0.804i)T \)
5 \( 1 + (-2.20 - 0.375i)T \)
7 \( 1 + (2.41 + 1.08i)T \)
good3 \( 1 + (3.24 - 0.614i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (0.382 - 0.975i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (2.24 + 0.253i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (4.74 + 0.177i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (2.27 - 3.93i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.113 + 3.03i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (8.85 - 2.02i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (6.89 - 3.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.30 - 2.27i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (0.461 + 0.957i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (0.129 + 0.368i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (-3.94 - 2.91i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (2.87 + 1.51i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (0.518 + 6.91i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-2.17 + 7.04i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (-4.14 - 1.11i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.26 + 14.2i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-3.74 + 2.76i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (9.48 + 5.47i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.70 - 15.1i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (-0.838 - 2.13i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (1.91 + 1.91i)T + 97iT^{2} \)
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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

−10.95710419647382702906380919462, −10.73481760704999189004745428005, −9.864629039298830051040011599630, −9.296459355498476477979347127955, −7.04928602137702851057486055292, −6.45845889653741820864373598294, −5.61296222621805964923790759418, −4.82907040139782612794154920736, −3.71936486437706714616479538832, −1.86626678791542545148054126451, 0.01598768821791593645831947422, 2.18170299968513009897507539010, 4.23676228237953132615123130800, 5.44157066789915017186927774302, 5.74642627727363389863215142980, 6.71125594192744666221207727405, 7.23288633341825468177645735823, 8.993920094225743901477588420087, 9.774000946364286850987621524931, 10.82308852019365098558715952114

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