Properties

Label 2-490-245.222-c1-0-9
Degree $2$
Conductor $490$
Sign $0.571 - 0.820i$
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  + (−0.757 + 0.652i)2-s + (2.51 + 1.33i)3-s + (0.149 − 0.988i)4-s + (−2.09 − 0.786i)5-s + (−2.77 + 0.633i)6-s + (1.67 − 2.04i)7-s + (0.532 + 0.846i)8-s + (2.87 + 4.21i)9-s + (2.09 − 0.769i)10-s + (−1.94 − 1.32i)11-s + (1.69 − 2.29i)12-s + (5.94 + 2.08i)13-s + (0.0627 + 2.64i)14-s + (−4.22 − 4.76i)15-s + (−0.955 − 0.294i)16-s + (3.16 + 7.25i)17-s + ⋯
L(s)  = 1  + (−0.535 + 0.461i)2-s + (1.45 + 0.768i)3-s + (0.0745 − 0.494i)4-s + (−0.936 − 0.351i)5-s + (−1.13 + 0.258i)6-s + (0.634 − 0.773i)7-s + (0.188 + 0.299i)8-s + (0.958 + 1.40i)9-s + (0.663 − 0.243i)10-s + (−0.586 − 0.399i)11-s + (0.488 − 0.661i)12-s + (1.64 + 0.576i)13-s + (0.0167 + 0.706i)14-s + (−1.09 − 1.23i)15-s + (−0.238 − 0.0736i)16-s + (0.767 + 1.75i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.571 - 0.820i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.571 - 0.820i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48774 + 0.777206i\)
\(L(\frac12)\) \(\approx\) \(1.48774 + 0.777206i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.757 - 0.652i)T \)
5 \( 1 + (2.09 + 0.786i)T \)
7 \( 1 + (-1.67 + 2.04i)T \)
good3 \( 1 + (-2.51 - 1.33i)T + (1.68 + 2.47i)T^{2} \)
11 \( 1 + (1.94 + 1.32i)T + (4.01 + 10.2i)T^{2} \)
13 \( 1 + (-5.94 - 2.08i)T + (10.1 + 8.10i)T^{2} \)
17 \( 1 + (-3.16 - 7.25i)T + (-11.5 + 12.4i)T^{2} \)
19 \( 1 + (0.174 - 0.301i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.01 - 1.31i)T + (15.6 + 16.8i)T^{2} \)
29 \( 1 + (0.208 - 0.166i)T + (6.45 - 28.2i)T^{2} \)
31 \( 1 + (-4.74 + 2.74i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.88 + 5.08i)T + (10.9 + 35.3i)T^{2} \)
41 \( 1 + (6.98 + 1.59i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (7.00 + 4.40i)T + (18.6 + 38.7i)T^{2} \)
47 \( 1 + (6.19 + 7.20i)T + (-7.00 + 46.4i)T^{2} \)
53 \( 1 + (-5.34 + 3.94i)T + (15.6 - 50.6i)T^{2} \)
59 \( 1 + (-0.862 + 0.800i)T + (4.40 - 58.8i)T^{2} \)
61 \( 1 + (-0.622 - 4.12i)T + (-58.2 + 17.9i)T^{2} \)
67 \( 1 + (0.871 - 3.25i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (5.89 - 7.39i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-1.49 + 1.73i)T + (-10.8 - 72.1i)T^{2} \)
79 \( 1 + (-6.39 - 3.69i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.53 + 7.24i)T + (-64.8 + 51.7i)T^{2} \)
89 \( 1 + (0.115 - 0.0790i)T + (32.5 - 82.8i)T^{2} \)
97 \( 1 + (13.0 - 13.0i)T - 97iT^{2} \)
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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

−10.68440754609901108005131336427, −10.23004391890680683779057468338, −8.889844905929286987021878081582, −8.320773343968601004807584651673, −8.054844663418188262730466978775, −6.87280773175043069244469663873, −5.29179041932338638186224877356, −3.98304235774261853661432051313, −3.55907500713204140646024352904, −1.53746080801273390687525325061, 1.33805776600053836306055323521, 2.84437540904099198799036341122, 3.26323850579941258081032235448, 4.89541080803063206032629128021, 6.69873638048301582864253088846, 7.60132173738970672041162085104, 8.288168148126726243829501512415, 8.656983284888421145508511363586, 9.754891562451928489338957199351, 10.89475982257650783848561278439

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