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Properties

Label 2-22e2-11.5-c1-0-8
Degree $2$
Conductor $484$
Sign $-0.923 - 0.382i$
Analytic cond. $3.86475$
Root an. cond. $1.96589$
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  + (−1.61 − 1.17i)3-s + (0.927 − 2.85i)5-s + (−2.80 + 2.03i)7-s + (0.309 + 0.951i)9-s + (−1.60 − 4.94i)13-s + (−4.85 + 3.52i)15-s + (−1.60 + 4.94i)17-s + (2.80 + 2.03i)19-s + 6.92·21-s − 6·23-s + (−3.23 − 2.35i)25-s + (−1.23 + 3.80i)27-s + (−4.20 + 3.05i)29-s + (−0.618 − 1.90i)31-s + (3.21 + 9.88i)35-s + ⋯
L(s)  = 1  + (−0.934 − 0.678i)3-s + (0.414 − 1.27i)5-s + (−1.05 + 0.769i)7-s + (0.103 + 0.317i)9-s + (−0.445 − 1.37i)13-s + (−1.25 + 0.910i)15-s + (−0.389 + 1.19i)17-s + (0.642 + 0.467i)19-s + 1.51·21-s − 1.25·23-s + (−0.647 − 0.470i)25-s + (−0.237 + 0.732i)27-s + (−0.780 + 0.567i)29-s + (−0.111 − 0.341i)31-s + (0.542 + 1.67i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(3.86475\)
Root analytic conductor: \(1.96589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{484} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 484,\ (\ :1/2),\ -0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0604199 + 0.303814i\)
\(L(\frac12)\) \(\approx\) \(0.0604199 + 0.303814i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (1.61 + 1.17i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.927 + 2.85i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.80 - 2.03i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.60 + 4.94i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.60 - 4.94i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.80 - 2.03i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (4.20 - 3.05i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.618 + 1.90i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.20 + 3.05i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (4.85 + 3.52i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.927 + 2.85i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.28 + 13.1i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.60 - 4.07i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.07 + 3.29i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.21 + 9.88i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (-1.54 - 4.75i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.39788479702593301514626756200, −9.609462526287078077681021394385, −8.711630832486617976719202133552, −7.77476392982939145943233378261, −6.43833957981006498421636928013, −5.75154886429547161638976765691, −5.19749228203966429147632901802, −3.50215957875391941378458216914, −1.76869752696301353148318202137, −0.19960093368595069560867127214, 2.49766105159827050397086796122, 3.76149481329744255786789402293, 4.81939614260779299824199944193, 6.07306447033311443365389805629, 6.75383275239340257222407988549, 7.44030760034659742960175535883, 9.325139882802923333900661599266, 9.888835175062593625607251479842, 10.50558850559596196672135891146, 11.40504034605511413650087728370

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