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Properties

Label 2-22e2-11.2-c0-0-0
Degree $2$
Conductor $484$
Sign $0.624 + 0.781i$
Analytic cond. $0.241547$
Root an. cond. $0.491474$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)3-s + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)15-s − 23-s + (0.309 + 0.951i)27-s + (−0.309 + 0.951i)31-s + (0.809 + 0.587i)37-s + (−1.61 + 1.17i)47-s + (0.309 + 0.951i)49-s + (0.618 − 1.90i)53-s + (0.809 + 0.587i)59-s − 67-s + (−0.809 + 0.587i)69-s + (−0.309 − 0.951i)71-s + (0.809 + 0.587i)81-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)15-s − 23-s + (0.309 + 0.951i)27-s + (−0.309 + 0.951i)31-s + (0.809 + 0.587i)37-s + (−1.61 + 1.17i)47-s + (0.309 + 0.951i)49-s + (0.618 − 1.90i)53-s + (0.809 + 0.587i)59-s − 67-s + (−0.809 + 0.587i)69-s + (−0.309 − 0.951i)71-s + (0.809 + 0.587i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $0.624 + 0.781i$
Analytic conductor: \(0.241547\)
Root analytic conductor: \(0.491474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{484} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 484,\ (\ :0),\ 0.624 + 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.037030106\)
\(L(\frac12)\) \(\approx\) \(1.037030106\)
\(L(1)\) 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 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
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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

−11.17637031849849935147176678074, −10.03175147461681629146812041804, −9.030282696148568188950768315256, −8.306740321877102695000451131511, −7.73526858035547090368709654650, −6.60722547573835642234384507499, −5.31408309055559401570823484682, −4.28718091686591138224921045754, −2.94323088331495041884507112057, −1.57343217524885922987096896120, 2.41862904401474906698208960983, 3.45010524351650637031987188390, 4.25486480901952770674551518193, 5.78023920827501190042452897403, 6.83836699581355881259483301235, 7.79809657383952038070420985908, 8.668229712712516547890205458938, 9.628471113452922889809490466949, 10.28748338170048509670783570699, 11.24233121575506221599441904429

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