Properties

Label 2-490-245.223-c1-0-25
Degree $2$
Conductor $490$
Sign $-0.910 + 0.413i$
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  + (0.330 − 0.943i)2-s + (−0.362 − 0.577i)3-s + (−0.781 − 0.623i)4-s + (1.73 − 1.41i)5-s + (−0.664 + 0.151i)6-s + (−2.08 − 1.62i)7-s + (−0.846 + 0.532i)8-s + (1.10 − 2.28i)9-s + (−0.764 − 2.10i)10-s + (−0.921 + 0.443i)11-s + (−0.0763 + 0.677i)12-s + (−0.713 + 2.03i)13-s + (−2.22 + 1.43i)14-s + (−1.44 − 0.485i)15-s + (0.222 + 0.974i)16-s + (0.178 − 1.58i)17-s + ⋯
L(s)  = 1  + (0.233 − 0.667i)2-s + (−0.209 − 0.333i)3-s + (−0.390 − 0.311i)4-s + (0.773 − 0.633i)5-s + (−0.271 + 0.0619i)6-s + (−0.788 − 0.614i)7-s + (−0.299 + 0.188i)8-s + (0.366 − 0.761i)9-s + (−0.241 − 0.664i)10-s + (−0.277 + 0.133i)11-s + (−0.0220 + 0.195i)12-s + (−0.197 + 0.565i)13-s + (−0.594 + 0.382i)14-s + (−0.373 − 0.125i)15-s + (0.0556 + 0.243i)16-s + (0.0432 − 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.910 + 0.413i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.910 + 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278174 - 1.28383i\)
\(L(\frac12)\) \(\approx\) \(0.278174 - 1.28383i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.330 + 0.943i)T \)
5 \( 1 + (-1.73 + 1.41i)T \)
7 \( 1 + (2.08 + 1.62i)T \)
good3 \( 1 + (0.362 + 0.577i)T + (-1.30 + 2.70i)T^{2} \)
11 \( 1 + (0.921 - 0.443i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.713 - 2.03i)T + (-10.1 - 8.10i)T^{2} \)
17 \( 1 + (-0.178 + 1.58i)T + (-16.5 - 3.78i)T^{2} \)
19 \( 1 - 0.760T + 19T^{2} \)
23 \( 1 + (3.53 - 0.398i)T + (22.4 - 5.11i)T^{2} \)
29 \( 1 + (-0.279 + 0.222i)T + (6.45 - 28.2i)T^{2} \)
31 \( 1 + 6.31iT - 31T^{2} \)
37 \( 1 + (-2.16 - 0.244i)T + (36.0 + 8.23i)T^{2} \)
41 \( 1 + (-2.96 - 0.677i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (0.499 - 0.795i)T + (-18.6 - 38.7i)T^{2} \)
47 \( 1 + (-1.21 - 0.423i)T + (36.7 + 29.3i)T^{2} \)
53 \( 1 + (6.84 - 0.771i)T + (51.6 - 11.7i)T^{2} \)
59 \( 1 + (-1.52 - 6.69i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-10.6 + 8.53i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (4.39 - 4.39i)T - 67iT^{2} \)
71 \( 1 + (-10.4 + 13.1i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-13.6 + 4.77i)T + (57.0 - 45.5i)T^{2} \)
79 \( 1 - 2.99iT - 79T^{2} \)
83 \( 1 + (-2.95 + 1.03i)T + (64.8 - 51.7i)T^{2} \)
89 \( 1 + (-10.6 - 5.14i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-4.64 - 4.64i)T + 97iT^{2} \)
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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

−10.49148098327349810456736231663, −9.557022729108196448460827945013, −9.368859483727875022167118840944, −7.84627263358286330443929164970, −6.65568793936678301652438500367, −5.92607181831084299810210694093, −4.69063699574623976874537371798, −3.67290924799009907795330906278, −2.18833920664267517923171005572, −0.75547458424188438274164147745, 2.35305412522409374424216065564, 3.54387684036920776798071537125, 5.03754467369599792167198191884, 5.75865248085453978607198304176, 6.58357529606097040822110259142, 7.58266167167237112636204435454, 8.618549688104432216304268284511, 9.750665038531930520251335376844, 10.22440457727034440934343422708, 11.18307062439287800032659966252

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