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Properties

Label 2-22e2-11.5-c1-0-4
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: $\chi_{484} (269, \cdot )$
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

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

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