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Properties

Label 2-490-35.17-c1-0-2
Degree $2$
Conductor $490$
Sign $-0.340 - 0.940i$
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  + (−0.258 + 0.965i)2-s + (−1.13 + 0.304i)3-s + (−0.866 − 0.499i)4-s + (−0.264 − 2.22i)5-s − 1.17i·6-s + (0.707 − 0.707i)8-s + (−1.40 + 0.810i)9-s + (2.21 + 0.318i)10-s + (−0.371 + 0.643i)11-s + (1.13 + 0.304i)12-s + (2.05 + 2.05i)13-s + (0.975 + 2.43i)15-s + (0.500 + 0.866i)16-s + (1.69 + 6.33i)17-s + (−0.419 − 1.56i)18-s + (0.946 + 1.63i)19-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.655 + 0.175i)3-s + (−0.433 − 0.249i)4-s + (−0.118 − 0.992i)5-s − 0.479i·6-s + (0.249 − 0.249i)8-s + (−0.467 + 0.270i)9-s + (0.699 + 0.100i)10-s + (−0.112 + 0.194i)11-s + (0.327 + 0.0877i)12-s + (0.570 + 0.570i)13-s + (0.251 + 0.629i)15-s + (0.125 + 0.216i)16-s + (0.411 + 1.53i)17-s + (−0.0988 − 0.368i)18-s + (0.217 + 0.375i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.340 - 0.940i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.340 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.437457 + 0.623493i\)
\(L(\frac12)\) \(\approx\) \(0.437457 + 0.623493i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (0.264 + 2.22i)T \)
7 \( 1 \)
good3 \( 1 + (1.13 - 0.304i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.371 - 0.643i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.05 - 2.05i)T + 13iT^{2} \)
17 \( 1 + (-1.69 - 6.33i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.946 - 1.63i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.11 - 1.36i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 9.69iT - 29T^{2} \)
31 \( 1 + (2.96 + 1.71i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.691 - 2.58i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.817iT - 41T^{2} \)
43 \( 1 + (-1.59 + 1.59i)T - 43iT^{2} \)
47 \( 1 + (4.54 + 1.21i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.29 - 4.81i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.27 - 2.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.25 - 3.03i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.2 + 3.54i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 + (-8.54 + 2.29i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.70 + 3.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.23 - 9.23i)T + 83iT^{2} \)
89 \( 1 + (-3.01 - 5.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.16 + 3.16i)T - 97iT^{2} \)
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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

−11.11433705278763805874616325040, −10.39118285607938660118968115463, −9.189102182566805515706807844856, −8.568871762540676182056691337626, −7.70599194155139698534395953362, −6.47293102833769585224275138333, −5.57192864653377217586447195453, −4.90320053450669065751380734654, −3.71782550785186712175764905537, −1.40851127730681802581966609545, 0.58339801152231673491198152772, 2.65046217030696564620882352851, 3.45815204695047252216405449630, 5.01190548817309536849232336084, 6.01749578257344220653818996964, 7.00590172595004819539117982203, 7.958567925444316087554903916421, 9.116910120721575932621341427985, 9.988355854934566877516991559234, 11.01033541734573233555938171345

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