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Properties

Label 2-490-245.239-c1-0-16
Degree $2$
Conductor $490$
Sign $0.954 + 0.298i$
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 + (−0.609 + 1.26i)3-s + (0.222 + 0.974i)4-s + (2.05 − 0.887i)5-s + (1.26 − 0.609i)6-s + (2.32 − 1.25i)7-s + (0.433 − 0.900i)8-s + (0.639 + 0.801i)9-s + (−2.15 − 0.585i)10-s + (0.870 − 1.09i)11-s + (−1.36 − 0.312i)12-s + (−4.70 − 3.75i)13-s + (−2.60 − 0.471i)14-s + (−0.127 + 3.13i)15-s + (−0.900 + 0.433i)16-s + (5.50 + 1.25i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (−0.352 + 0.730i)3-s + (0.111 + 0.487i)4-s + (0.917 − 0.397i)5-s + (0.516 − 0.248i)6-s + (0.880 − 0.474i)7-s + (0.153 − 0.318i)8-s + (0.213 + 0.267i)9-s + (−0.682 − 0.185i)10-s + (0.262 − 0.329i)11-s + (−0.395 − 0.0902i)12-s + (−1.30 − 1.04i)13-s + (−0.695 − 0.126i)14-s + (−0.0328 + 0.810i)15-s + (−0.225 + 0.108i)16-s + (1.33 + 0.304i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.954 + 0.298i$
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: \((2,\ 490,\ (\ :1/2),\ 0.954 + 0.298i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24405 - 0.189780i\)
\(L(\frac12)\) \(\approx\) \(1.24405 - 0.189780i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.05 + 0.887i)T \)
7 \( 1 + (-2.32 + 1.25i)T \)
good3 \( 1 + (0.609 - 1.26i)T + (-1.87 - 2.34i)T^{2} \)
11 \( 1 + (-0.870 + 1.09i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (4.70 + 3.75i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-5.50 - 1.25i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 + 0.518T + 19T^{2} \)
23 \( 1 + (3.76 - 0.858i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.815 + 3.57i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 9.05T + 31T^{2} \)
37 \( 1 + (-9.64 - 2.20i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + (-1.08 - 0.522i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-3.28 - 6.82i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (2.36 + 1.88i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (8.79 - 2.00i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (3.85 - 1.85i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-0.387 + 1.69i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 12.7iT - 67T^{2} \)
71 \( 1 + (0.131 + 0.576i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-10.5 + 8.43i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + (0.153 - 0.122i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (-4.00 - 5.02i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 9.57iT - 97T^{2} \)
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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

−10.63515478523729220919972475230, −9.949727154073781804369350814409, −9.664415431410708269091317802289, −8.136145628658345076257841483613, −7.72235516737808029088154932475, −6.07908074519452410992701098167, −5.08989646241368500990467176055, −4.28861199850701087861020088977, −2.65807819154172844254211768906, −1.18936120702766934672273190169, 1.40383656342401187630930734987, 2.42178966866602851019291867357, 4.60772338263097580062733356905, 5.67136701635711695112281943933, 6.52179436824058675616906430326, 7.28125951198765749312851043119, 8.109840688423728350597380812945, 9.496822658285426581118145962365, 9.740537897424422059283690146904, 10.97336733822311032906947111483

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