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Properties

Label 2-490-5.4-c3-0-61
Degree $2$
Conductor $490$
Sign $0.905 - 0.425i$
Analytic cond. $28.9109$
Root an. cond. $5.37688$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 9.74i·3-s − 4·4-s + (−4.75 − 10.1i)5-s − 19.4·6-s + 8i·8-s − 67.9·9-s + (−20.2 + 9.51i)10-s + 6.52·11-s + 38.9i·12-s + 41.6i·13-s + (−98.6 + 46.3i)15-s + 16·16-s − 109. i·17-s + 135. i·18-s + 29.7·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.87i·3-s − 0.5·4-s + (−0.425 − 0.905i)5-s − 1.32·6-s + 0.353i·8-s − 2.51·9-s + (−0.639 + 0.300i)10-s + 0.178·11-s + 0.937i·12-s + 0.888i·13-s + (−1.69 + 0.797i)15-s + 0.250·16-s − 1.56i·17-s + 1.78i·18-s + 0.359·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.425i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.905 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.905 - 0.425i$
Analytic conductor: \(28.9109\)
Root analytic conductor: \(5.37688\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :3/2),\ 0.905 - 0.425i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3587897380\)
\(L(\frac12)\) \(\approx\) \(0.3587897380\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
5 \( 1 + (4.75 + 10.1i)T \)
7 \( 1 \)
good3 \( 1 + 9.74iT - 27T^{2} \)
11 \( 1 - 6.52T + 1.33e3T^{2} \)
13 \( 1 - 41.6iT - 2.19e3T^{2} \)
17 \( 1 + 109. iT - 4.91e3T^{2} \)
19 \( 1 - 29.7T + 6.85e3T^{2} \)
23 \( 1 - 180. iT - 1.21e4T^{2} \)
29 \( 1 + 183.T + 2.43e4T^{2} \)
31 \( 1 - 116.T + 2.97e4T^{2} \)
37 \( 1 + 396. iT - 5.06e4T^{2} \)
41 \( 1 + 197.T + 6.89e4T^{2} \)
43 \( 1 + 302. iT - 7.95e4T^{2} \)
47 \( 1 - 277. iT - 1.03e5T^{2} \)
53 \( 1 - 405. iT - 1.48e5T^{2} \)
59 \( 1 + 67.8T + 2.05e5T^{2} \)
61 \( 1 + 510.T + 2.26e5T^{2} \)
67 \( 1 + 1.02e3iT - 3.00e5T^{2} \)
71 \( 1 + 352.T + 3.57e5T^{2} \)
73 \( 1 + 59.7iT - 3.89e5T^{2} \)
79 \( 1 - 133.T + 4.93e5T^{2} \)
83 \( 1 + 571. iT - 5.71e5T^{2} \)
89 \( 1 - 547.T + 7.04e5T^{2} \)
97 \( 1 + 783. iT - 9.12e5T^{2} \)
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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

−9.305069117661445509415791672454, −8.984604182686364566317715259433, −7.63384294308156504564884266401, −7.35019520125303305466266586734, −5.95588111828386732589757978200, −4.97724213882859631443995602340, −3.46716939096674654092375483471, −2.09741316118424930975066944658, −1.19328236535738638054366556766, −0.12264614034763938718455466265, 2.96381007691675022961302665511, 3.81097616264180980857035062942, 4.65959755702189590227768481945, 5.74949955773848929005328353603, 6.57978169193039772203146106252, 8.088223333868553287411987240326, 8.556988319793063653902178881166, 9.833689278338293009395803589877, 10.31907363840352090287496092028, 10.98388007219051171154100557099

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