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Properties

Label 2-490-245.239-c1-0-19
Degree $2$
Conductor $490$
Sign $0.957 - 0.287i$
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 + (0.534 − 1.10i)3-s + (0.222 + 0.974i)4-s + (2.12 − 0.693i)5-s + (1.10 − 0.534i)6-s + (1.96 + 1.77i)7-s + (−0.433 + 0.900i)8-s + (0.925 + 1.16i)9-s + (2.09 + 0.782i)10-s + (−0.144 + 0.180i)11-s + (1.20 + 0.273i)12-s + (−2.82 − 2.25i)13-s + (0.432 + 2.61i)14-s + (0.366 − 2.72i)15-s + (−0.900 + 0.433i)16-s + (−4.28 − 0.978i)17-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (0.308 − 0.640i)3-s + (0.111 + 0.487i)4-s + (0.950 − 0.310i)5-s + (0.452 − 0.218i)6-s + (0.742 + 0.669i)7-s + (−0.153 + 0.318i)8-s + (0.308 + 0.386i)9-s + (0.662 + 0.247i)10-s + (−0.0435 + 0.0545i)11-s + (0.346 + 0.0790i)12-s + (−0.784 − 0.625i)13-s + (0.115 + 0.697i)14-s + (0.0945 − 0.704i)15-s + (−0.225 + 0.108i)16-s + (−1.04 − 0.237i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.957 - 0.287i$
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: \((2,\ 490,\ (\ :1/2),\ 0.957 - 0.287i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48486 + 0.364367i\)
\(L(\frac12)\) \(\approx\) \(2.48486 + 0.364367i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 0.693i)T \)
7 \( 1 + (-1.96 - 1.77i)T \)
good3 \( 1 + (-0.534 + 1.10i)T + (-1.87 - 2.34i)T^{2} \)
11 \( 1 + (0.144 - 0.180i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (2.82 + 2.25i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (4.28 + 0.978i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 + 5.37T + 19T^{2} \)
23 \( 1 + (-6.14 + 1.40i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (1.35 - 5.91i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 2.69T + 31T^{2} \)
37 \( 1 + (-8.34 - 1.90i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + (9.33 + 4.49i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (3.98 + 8.26i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (7.27 + 5.80i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (9.43 - 2.15i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-8.60 + 4.14i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (1.02 - 4.51i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + 3.08iT - 67T^{2} \)
71 \( 1 + (0.708 + 3.10i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (3.09 - 2.46i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 8.99T + 79T^{2} \)
83 \( 1 + (7.40 - 5.90i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (6.17 + 7.73i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 8.76iT - 97T^{2} \)
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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

−11.06979413350346706364636294377, −10.14131617641027842797466303548, −8.826974678146084774172478369686, −8.367322793577806231810862460140, −7.18830215182844339805195677885, −6.44890503658457972472300504776, −5.17917443424892925244085380995, −4.73157489213639693139334584637, −2.69662722559520879486386331043, −1.86021859969043613350563709635, 1.68344251751476288051013829249, 2.90452539952185175022856501113, 4.33911808344048678700163348829, 4.77152367746940438616780912960, 6.27479866485014626974042380975, 6.98271512108547409693372338210, 8.438014067376274591448306810485, 9.523541382145188366690535475800, 9.996589947007169715607428989594, 10.96144468563905289564248809545

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