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Properties

Label 2-22e2-11.5-c1-0-2
Degree $2$
Conductor $484$
Sign $-0.353 - 0.935i$
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  + (2.11 + 1.53i)3-s + (−0.5 + 1.53i)5-s + (−3.11 + 2.26i)7-s + (1.19 + 3.66i)9-s + (−0.736 − 2.26i)13-s + (−3.42 + 2.48i)15-s + (−0.736 + 2.26i)17-s + (3.11 + 2.26i)19-s − 10.0·21-s − 2.47·23-s + (1.92 + 1.40i)25-s + (−0.690 + 2.12i)27-s + (6.97 − 5.06i)29-s + (0.263 + 0.812i)31-s + (−1.92 − 5.93i)35-s + ⋯
L(s)  = 1  + (1.22 + 0.888i)3-s + (−0.223 + 0.688i)5-s + (−1.17 + 0.856i)7-s + (0.396 + 1.22i)9-s + (−0.204 − 0.628i)13-s + (−0.884 + 0.642i)15-s + (−0.178 + 0.549i)17-s + (0.715 + 0.519i)19-s − 2.20·21-s − 0.515·23-s + (0.385 + 0.280i)25-s + (−0.132 + 0.409i)27-s + (1.29 − 0.940i)29-s + (0.0474 + 0.145i)31-s + (−0.325 − 1.00i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $-0.353 - 0.935i$
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: \((2,\ 484,\ (\ :1/2),\ -0.353 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.970331 + 1.40458i\)
\(L(\frac12)\) \(\approx\) \(0.970331 + 1.40458i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.11 - 1.53i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.5 - 1.53i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (3.11 - 2.26i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.736 + 2.26i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.736 - 2.26i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.11 - 2.26i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + (-6.97 + 5.06i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.263 - 0.812i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.5 + 1.08i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.97 - 5.06i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (1.11 + 0.812i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.26 + 3.88i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.881 - 0.640i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.736 - 2.26i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + (1.97 - 6.06i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.736 - 0.534i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.20 - 6.79i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.02 - 12.3i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 0.472T + 89T^{2} \)
97 \( 1 + (4.5 + 13.8i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.02917448605377372217499598198, −9.900331967279332693688922653910, −9.741602811821488886616160054746, −8.612629150638871374905518245930, −7.921839358720243012940176755082, −6.66623035760577366553208148121, −5.64862261427245014913852925531, −4.16397685369976512602406170840, −3.16622664921507288412484857846, −2.61576590852198719548922799088, 0.932517388457358335170973452081, 2.57122579690891634744221049847, 3.58037246900776673949149662476, 4.75742052953763658760168065716, 6.47374851336202765233537204903, 7.14127905763164215739289470271, 7.937261748050835257795376719047, 8.969270491426240862613964243295, 9.458739195022888990089187601324, 10.55196343140305410649084066623

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