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Properties

Label 2-490-5.3-c2-0-28
Degree $2$
Conductor $490$
Sign $0.229 + 0.973i$
Analytic cond. $13.3515$
Root an. cond. $3.65397$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (2 + 2i)3-s − 2i·4-s − 5i·5-s − 4·6-s + (2 + 2i)8-s − i·9-s + (5 + 5i)10-s − 8·11-s + (4 − 4i)12-s + (−3 − 3i)13-s + (10 − 10i)15-s − 4·16-s + (−7 + 7i)17-s + (1 + i)18-s − 20i·19-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.666 + 0.666i)3-s − 0.5i·4-s − i·5-s − 0.666·6-s + (0.250 + 0.250i)8-s − 0.111i·9-s + (0.5 + 0.5i)10-s − 0.727·11-s + (0.333 − 0.333i)12-s + (−0.230 − 0.230i)13-s + (0.666 − 0.666i)15-s − 0.250·16-s + (−0.411 + 0.411i)17-s + (0.0555 + 0.0555i)18-s − 1.05i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(13.3515\)
Root analytic conductor: \(3.65397\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.001690440\)
\(L(\frac12)\) \(\approx\) \(1.001690440\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
5 \( 1 + 5iT \)
7 \( 1 \)
good3 \( 1 + (-2 - 2i)T + 9iT^{2} \)
11 \( 1 + 8T + 121T^{2} \)
13 \( 1 + (3 + 3i)T + 169iT^{2} \)
17 \( 1 + (7 - 7i)T - 289iT^{2} \)
19 \( 1 + 20iT - 361T^{2} \)
23 \( 1 + (2 + 2i)T + 529iT^{2} \)
29 \( 1 + 40iT - 841T^{2} \)
31 \( 1 + 52T + 961T^{2} \)
37 \( 1 + (3 - 3i)T - 1.36e3iT^{2} \)
41 \( 1 - 8T + 1.68e3T^{2} \)
43 \( 1 + (42 + 42i)T + 1.84e3iT^{2} \)
47 \( 1 + (-18 + 18i)T - 2.20e3iT^{2} \)
53 \( 1 + (-53 - 53i)T + 2.80e3iT^{2} \)
59 \( 1 + 20iT - 3.48e3T^{2} \)
61 \( 1 - 48T + 3.72e3T^{2} \)
67 \( 1 + (-62 + 62i)T - 4.48e3iT^{2} \)
71 \( 1 + 28T + 5.04e3T^{2} \)
73 \( 1 + (-47 - 47i)T + 5.32e3iT^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + (18 + 18i)T + 6.88e3iT^{2} \)
89 \( 1 - 80iT - 7.92e3T^{2} \)
97 \( 1 + (-63 + 63i)T - 9.40e3iT^{2} \)
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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

−10.20472395532578646125634402473, −9.477031595699162256955455371676, −8.776120356321155462373302008104, −8.155174804634571168044242753486, −7.09695541464832243307087124796, −5.78203109036303042538656275326, −4.85563381940162307957273698765, −3.84380557111556049011140733996, −2.27875040427668673378062811026, −0.41547412212647844524354877098, 1.77701252894254841761466145373, 2.66465637967517539330946638020, 3.66002745272228595442641070965, 5.31053474825517591672987491799, 6.76844036089540480799115309114, 7.45198018515952107404748905383, 8.157451646993707909230284565480, 9.141268292069782617519987736448, 10.18098052018954231364600002650, 10.81723544830392380277792708597

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