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Properties

Label 2-490-245.103-c1-0-12
Degree $2$
Conductor $490$
Sign $0.961 + 0.276i$
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  + (−0.593 + 0.804i)2-s + (−0.454 + 0.0859i)3-s + (−0.294 − 0.955i)4-s + (−0.288 − 2.21i)5-s + (0.200 − 0.416i)6-s + (1.03 + 2.43i)7-s + (0.943 + 0.330i)8-s + (−2.59 + 1.01i)9-s + (1.95 + 1.08i)10-s + (0.944 − 2.40i)11-s + (0.216 + 0.408i)12-s + (−0.132 − 0.0149i)13-s + (−2.57 − 0.616i)14-s + (0.321 + 0.982i)15-s + (−0.826 + 0.563i)16-s + (6.79 + 0.254i)17-s + ⋯
L(s)  = 1  + (−0.419 + 0.568i)2-s + (−0.262 + 0.0496i)3-s + (−0.147 − 0.477i)4-s + (−0.129 − 0.991i)5-s + (0.0818 − 0.170i)6-s + (0.390 + 0.920i)7-s + (0.333 + 0.116i)8-s + (−0.864 + 0.339i)9-s + (0.618 + 0.342i)10-s + (0.284 − 0.725i)11-s + (0.0623 + 0.117i)12-s + (−0.0368 − 0.00415i)13-s + (−0.687 − 0.164i)14-s + (0.0830 + 0.253i)15-s + (−0.206 + 0.140i)16-s + (1.64 + 0.0616i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.961 + 0.276i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.961 + 0.276i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.980257 - 0.138029i\)
\(L(\frac12)\) \(\approx\) \(0.980257 - 0.138029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.593 - 0.804i)T \)
5 \( 1 + (0.288 + 2.21i)T \)
7 \( 1 + (-1.03 - 2.43i)T \)
good3 \( 1 + (0.454 - 0.0859i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (-0.944 + 2.40i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (0.132 + 0.0149i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (-6.79 - 0.254i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (-2.75 + 4.76i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.276 + 7.37i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (-5.15 + 1.17i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (1.16 - 0.675i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.80 + 4.65i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (1.12 + 2.34i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (1.26 + 3.62i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (-1.12 - 0.829i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (-5.68 - 3.00i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (-0.672 - 8.97i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (2.86 - 9.27i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (-2.99 - 0.802i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.632 + 2.77i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-0.198 + 0.146i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (14.5 + 8.37i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.09 - 9.69i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (4.22 + 10.7i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (3.45 + 3.45i)T + 97iT^{2} \)
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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

−10.96469925504302580344941885411, −9.826580756955916292948366463051, −8.719506554526182863995065212849, −8.539317326368866943131280735781, −7.48494006349062595302644899141, −5.96943267340844979603592402479, −5.51289041491541788469694848718, −4.54789495689481326682623668587, −2.73060000551329797173955142889, −0.839567030194068405475040390023, 1.33644832237104761929295005804, 3.06072602988760504591678714304, 3.84634003294707823758773823624, 5.35259004725420532767171139844, 6.53802267649025018394652874192, 7.58873550322365842977971045366, 8.068018374945860408583092175213, 9.750438391434608230513031106909, 9.969534572348625256793636028931, 11.09402456440585433397110035743

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