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Properties

Label 2-22e2-11.9-c1-0-8
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: $\chi_{484} (9, \cdot )$
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

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

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