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Properties

Label 2-22e2-11.4-c1-0-3
Degree $2$
Conductor $484$
Sign $0.898 - 0.437i$
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  + (0.618 − 1.90i)3-s + (−2.42 + 1.76i)5-s + (1.07 + 3.29i)7-s + (−0.809 − 0.587i)9-s + (4.20 + 3.05i)13-s + (1.85 + 5.70i)15-s + (4.20 − 3.05i)17-s + (−1.07 + 3.29i)19-s + 6.92·21-s − 6·23-s + (1.23 − 3.80i)25-s + (3.23 − 2.35i)27-s + (1.60 + 4.94i)29-s + (1.61 + 1.17i)31-s + (−8.40 − 6.10i)35-s + ⋯
L(s)  = 1  + (0.356 − 1.09i)3-s + (−1.08 + 0.788i)5-s + (0.404 + 1.24i)7-s + (−0.269 − 0.195i)9-s + (1.16 + 0.847i)13-s + (0.478 + 1.47i)15-s + (1.01 − 0.740i)17-s + (−0.245 + 0.755i)19-s + 1.51·21-s − 1.25·23-s + (0.247 − 0.760i)25-s + (0.622 − 0.452i)27-s + (0.298 + 0.917i)29-s + (0.290 + 0.211i)31-s + (−1.42 − 1.03i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $0.898 - 0.437i$
Analytic conductor: \(3.86475\)
Root analytic conductor: \(1.96589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{484} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 484,\ (\ :1/2),\ 0.898 - 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37035 + 0.316062i\)
\(L(\frac12)\) \(\approx\) \(1.37035 + 0.316062i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.618 + 1.90i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (2.42 - 1.76i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.07 - 3.29i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.20 - 3.05i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.20 + 3.05i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.07 - 3.29i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (-1.60 - 4.94i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.61 - 1.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.60 + 4.94i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-1.85 + 5.70i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.42 - 1.76i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (11.2 - 8.14i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.14 - 6.58i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.80 - 2.03i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.40 - 6.10i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (4.04 + 2.93i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.34539378763698768062730011466, −10.28406838221100963188543133413, −8.881666859338304905959545279564, −8.205611213768174939401269582332, −7.49158919757557723340443892674, −6.61930686444053648316443566911, −5.63592215650930187885469294907, −4.04007893770970147807582721753, −2.88227273536465605248164758697, −1.65481890429106589581046978217, 0.932800103831873987413416072526, 3.45634746713046901247808587247, 4.08461867066035985153050145492, 4.74379905838905488574864231847, 6.16115110607912616539623375535, 7.79788212990948084812626670681, 8.064613599416460806924910895324, 9.110364935955963222628214807770, 10.20153258707320922605263001091, 10.69722164062410384369575536918

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