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Properties

Label 2-490-245.103-c1-0-14
Degree $2$
Conductor $490$
Sign $0.846 - 0.532i$
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 + (2.06 − 0.389i)3-s + (−0.294 − 0.955i)4-s + (−0.497 + 2.17i)5-s + (−0.910 + 1.88i)6-s + (1.43 − 2.22i)7-s + (0.943 + 0.330i)8-s + (1.30 − 0.511i)9-s + (−1.45 − 1.69i)10-s + (1.09 − 2.78i)11-s + (−0.980 − 1.85i)12-s + (5.09 + 0.574i)13-s + (0.938 + 2.47i)14-s + (−0.175 + 4.68i)15-s + (−0.826 + 0.563i)16-s + (0.430 + 0.0160i)17-s + ⋯
L(s)  = 1  + (−0.419 + 0.568i)2-s + (1.18 − 0.225i)3-s + (−0.147 − 0.477i)4-s + (−0.222 + 0.974i)5-s + (−0.371 + 0.771i)6-s + (0.541 − 0.840i)7-s + (0.333 + 0.116i)8-s + (0.434 − 0.170i)9-s + (−0.461 − 0.535i)10-s + (0.329 − 0.839i)11-s + (−0.282 − 0.535i)12-s + (1.41 + 0.159i)13-s + (0.250 + 0.661i)14-s + (−0.0453 + 1.21i)15-s + (−0.206 + 0.140i)16-s + (0.104 + 0.00390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.846 - 0.532i$
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: \((2,\ 490,\ (\ :1/2),\ 0.846 - 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69024 + 0.487374i\)
\(L(\frac12)\) \(\approx\) \(1.69024 + 0.487374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.497 - 2.17i)T \)
7 \( 1 + (-1.43 + 2.22i)T \)
good3 \( 1 + (-2.06 + 0.389i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (-1.09 + 2.78i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (-5.09 - 0.574i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (-0.430 - 0.0160i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (-0.674 + 1.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.323 - 8.65i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (-7.47 + 1.70i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (8.14 - 4.70i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.67 + 4.05i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (-1.90 - 3.95i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (1.90 + 5.44i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (9.24 + 6.82i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (4.00 + 2.11i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (-1.08 - 14.4i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-1.44 + 4.68i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (2.33 + 0.626i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.922 - 4.04i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (1.03 - 0.760i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (1.12 + 0.650i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0299 + 0.265i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (3.51 + 8.94i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (1.55 + 1.55i)T + 97iT^{2} \)
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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

−11.06899828824561298646077366998, −10.01291552531835061076505194782, −8.969677317461442596551247153868, −8.251362474247989168613275490954, −7.55647106349898475936352350218, −6.77164700045516594843287806388, −5.68173612380918492028327242053, −3.92594884958553195857143495810, −3.18325485114113432572584713407, −1.49096020120218433832331679704, 1.47681915590481501345240859392, 2.66850523152209863465375837963, 3.91441721893593292864013633092, 4.80657256789918845627211126800, 6.25423771413473932474395128481, 8.005820828374075039451500679509, 8.267836300364942346266695751279, 9.100499431773837533473071131585, 9.607814319702631156396730320238, 10.87480673948570925011001151829

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