Properties

Label 2-22e2-121.100-c1-0-3
Degree $2$
Conductor $484$
Sign $0.374 - 0.927i$
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  − 0.390·3-s + (1.96 − 0.576i)5-s + (−0.526 + 3.65i)7-s − 2.84·9-s + (2.68 + 1.94i)11-s + (−4.47 + 5.16i)13-s + (−0.767 + 0.225i)15-s + (4.70 − 3.02i)17-s + (3.38 + 2.17i)19-s + (0.205 − 1.42i)21-s + (−0.357 − 2.48i)23-s + (−0.680 + 0.437i)25-s + 2.28·27-s + (−4.22 − 2.71i)29-s + (4.81 + 5.55i)31-s + ⋯
L(s)  = 1  − 0.225·3-s + (0.878 − 0.257i)5-s + (−0.198 + 1.38i)7-s − 0.949·9-s + (0.809 + 0.586i)11-s + (−1.24 + 1.43i)13-s + (−0.198 + 0.0581i)15-s + (1.14 − 0.732i)17-s + (0.777 + 0.499i)19-s + (0.0448 − 0.311i)21-s + (−0.0745 − 0.518i)23-s + (−0.136 + 0.0874i)25-s + 0.439·27-s + (−0.785 − 0.504i)29-s + (0.864 + 0.997i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(3.86475\)
Root analytic conductor: \(1.96589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{484} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 484,\ (\ :1/2),\ 0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10068 + 0.742299i\)
\(L(\frac12)\) \(\approx\) \(1.10068 + 0.742299i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.68 - 1.94i)T \)
good3 \( 1 + 0.390T + 3T^{2} \)
5 \( 1 + (-1.96 + 0.576i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (0.526 - 3.65i)T + (-6.71 - 1.97i)T^{2} \)
13 \( 1 + (4.47 - 5.16i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-4.70 + 3.02i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-3.38 - 2.17i)T + (7.89 + 17.2i)T^{2} \)
23 \( 1 + (0.357 + 2.48i)T + (-22.0 + 6.47i)T^{2} \)
29 \( 1 + (4.22 + 2.71i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-4.81 - 5.55i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (2.20 + 2.54i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (-5.18 - 11.3i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-5.07 - 1.49i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + (-1.72 + 3.76i)T + (-30.7 - 35.5i)T^{2} \)
53 \( 1 + (-0.659 + 4.58i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (0.325 - 0.712i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (0.440 - 0.963i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (2.04 + 4.48i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (1.87 + 1.20i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (2.25 + 15.6i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (6.55 - 1.92i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-0.413 + 2.87i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (3.21 - 2.06i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (2.35 + 0.692i)T + (81.6 + 52.4i)T^{2} \)
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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

−11.50919218105039568926820259086, −9.828837307569495785721472315872, −9.489088841206676445463040870316, −8.763875130011546432569479455848, −7.43939116418482654811475253904, −6.28289687200564523982515235929, −5.59000317098783256511337178100, −4.71119437683156271188138303631, −2.91749668816696841375754615344, −1.86640966862029136016733709865, 0.853900440263696063693794559654, 2.78226991980312964230984972624, 3.83274220963884978498180902084, 5.42653450896760999796882670734, 5.95174104346691633036343869908, 7.20481490593990754083549698778, 7.913628320203575000502148762334, 9.241837725504973643569649819229, 10.09051521587906351122520873878, 10.60712027572204728995067369422

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