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Properties

Label 2-22e2-11.9-c1-0-0
Degree $2$
Conductor $484$
Sign $-0.780 - 0.625i$
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  + (−0.809 + 0.587i)3-s + (−0.927 − 2.85i)5-s + (−1.61 − 1.17i)7-s + (−0.618 + 1.90i)9-s + (−1.23 + 3.80i)13-s + (2.42 + 1.76i)15-s + (1.85 + 5.70i)17-s + (−6.47 + 4.70i)19-s + 2·21-s − 3·23-s + (−3.23 + 2.35i)25-s + (−1.54 − 4.75i)27-s + (1.54 − 4.75i)31-s + (−1.85 + 5.70i)35-s + (0.809 + 0.587i)37-s + ⋯
L(s)  = 1  + (−0.467 + 0.339i)3-s + (−0.414 − 1.27i)5-s + (−0.611 − 0.444i)7-s + (−0.206 + 0.634i)9-s + (−0.342 + 1.05i)13-s + (0.626 + 0.455i)15-s + (0.449 + 1.38i)17-s + (−1.48 + 1.07i)19-s + 0.436·21-s − 0.625·23-s + (−0.647 + 0.470i)25-s + (−0.297 − 0.915i)27-s + (0.277 − 0.854i)31-s + (−0.313 + 0.964i)35-s + (0.133 + 0.0966i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $-0.780 - 0.625i$
Analytic conductor: \(3.86475\)
Root analytic conductor: \(1.96589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{484} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 484,\ (\ :1/2),\ -0.780 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0921696 + 0.262282i\)
\(L(\frac12)\) \(\approx\) \(0.0921696 + 0.262282i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 - 0.587i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.927 + 2.85i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (1.61 + 1.17i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.23 - 3.80i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.85 - 5.70i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (6.47 - 4.70i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.54 + 4.75i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.85 - 5.70i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.42 + 1.76i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.23 + 3.80i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + T + 67T^{2} \)
71 \( 1 + (-4.63 - 14.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.23 - 2.35i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.618 + 1.90i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.85 - 5.70i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (2.16 - 6.65i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.34970435553152935078958819434, −10.35545445782148698210673255615, −9.709933958167246672157861215042, −8.467634654084876319144300494686, −8.023052977711140106762878543094, −6.55711344404985079679555625142, −5.65970258519362774457983103592, −4.47776785017586715424293443886, −3.93681766421804760894024903056, −1.82402057370108355420747770955, 0.17005373709636390254425675217, 2.67592984715128118571751060804, 3.39073667817000371159085184578, 5.03782732549100163009015424012, 6.25677257572320353983489486094, 6.77251748149464519660111346696, 7.67388878203121218327849956485, 8.880993819863908299184910047812, 9.893426035425594820715802253990, 10.72597901343989641727364167919

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