Properties

Label 2-490-245.239-c1-0-17
Degree $2$
Conductor $490$
Sign $0.443 + 0.896i$
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 + (0.864 − 1.79i)3-s + (0.222 + 0.974i)4-s + (1.67 + 1.48i)5-s + (−1.79 + 0.864i)6-s + (2.09 − 1.61i)7-s + (0.433 − 0.900i)8-s + (−0.605 − 0.759i)9-s + (−0.383 − 2.20i)10-s + (−0.709 + 0.890i)11-s + (1.94 + 0.443i)12-s + (1.60 + 1.27i)13-s + (−2.64 − 0.0449i)14-s + (4.10 − 1.72i)15-s + (−0.900 + 0.433i)16-s + (−0.405 − 0.0925i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (0.499 − 1.03i)3-s + (0.111 + 0.487i)4-s + (0.748 + 0.663i)5-s + (−0.733 + 0.352i)6-s + (0.792 − 0.610i)7-s + (0.153 − 0.318i)8-s + (−0.201 − 0.253i)9-s + (−0.121 − 0.696i)10-s + (−0.214 + 0.268i)11-s + (0.560 + 0.128i)12-s + (0.444 + 0.354i)13-s + (−0.707 − 0.0120i)14-s + (1.06 − 0.444i)15-s + (−0.225 + 0.108i)16-s + (−0.0983 − 0.0224i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.443 + 0.896i$
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: \((2,\ 490,\ (\ :1/2),\ 0.443 + 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35335 - 0.840817i\)
\(L(\frac12)\) \(\approx\) \(1.35335 - 0.840817i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 - 1.48i)T \)
7 \( 1 + (-2.09 + 1.61i)T \)
good3 \( 1 + (-0.864 + 1.79i)T + (-1.87 - 2.34i)T^{2} \)
11 \( 1 + (0.709 - 0.890i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-1.60 - 1.27i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.405 + 0.0925i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 - 3.74T + 19T^{2} \)
23 \( 1 + (0.412 - 0.0942i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.276 + 1.20i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 2.98T + 31T^{2} \)
37 \( 1 + (2.53 + 0.577i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + (8.98 + 4.32i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (4.11 + 8.53i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (0.165 + 0.132i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (12.9 - 2.95i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-3.09 + 1.49i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (0.440 - 1.93i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + 4.24iT - 67T^{2} \)
71 \( 1 + (-1.57 - 6.91i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (4.40 - 3.50i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 + (4.28 - 3.41i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (-10.4 - 13.0i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 4.55iT - 97T^{2} \)
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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

−10.72828107432990021340182651238, −10.02314461848118253683660360497, −8.962619080993275268362732211306, −8.020397341252137963425320963874, −7.27385796936288881419009183431, −6.62510087606038831532450400905, −5.15104194077005023107915411753, −3.54996447265413219378055566640, −2.22962555760614314530902856984, −1.43644688910069177080755349174, 1.55844919212735858370637510150, 3.15400552788611562603602337153, 4.71486269762173148651909162255, 5.33543023199160535174960798246, 6.38861419903231691456557137571, 7.922220716016630205355711487327, 8.594322607394932829828148212988, 9.279548847121989374815548115249, 9.946459491144859371200704204040, 10.79776465080201794161571097854

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