Properties

Label 2-522-9.4-c1-0-7
Degree $2$
Conductor $522$
Sign $0.978 + 0.208i$
Analytic cond. $4.16819$
Root an. cond. $2.04161$
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  + (−0.5 − 0.866i)2-s + (−0.120 − 1.72i)3-s + (−0.499 + 0.866i)4-s + (−1.19 + 2.07i)5-s + (−1.43 + 0.968i)6-s + (0.279 + 0.483i)7-s + 0.999·8-s + (−2.97 + 0.416i)9-s + 2.39·10-s + (1.31 + 2.27i)11-s + (1.55 + 0.759i)12-s + (0.00438 − 0.00759i)13-s + (0.279 − 0.483i)14-s + (3.72 + 1.81i)15-s + (−0.5 − 0.866i)16-s + 3.44·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.0695 − 0.997i)3-s + (−0.249 + 0.433i)4-s + (−0.535 + 0.927i)5-s + (−0.586 + 0.395i)6-s + (0.105 + 0.182i)7-s + 0.353·8-s + (−0.990 + 0.138i)9-s + 0.756·10-s + (0.396 + 0.687i)11-s + (0.449 + 0.219i)12-s + (0.00121 − 0.00210i)13-s + (0.0745 − 0.129i)14-s + (0.962 + 0.469i)15-s + (−0.125 − 0.216i)16-s + 0.834·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(522\)    =    \(2 \cdot 3^{2} \cdot 29\)
Sign: $0.978 + 0.208i$
Analytic conductor: \(4.16819\)
Root analytic conductor: \(2.04161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{522} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 522,\ (\ :1/2),\ 0.978 + 0.208i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.997310 - 0.105013i\)
\(L(\frac12)\) \(\approx\) \(0.997310 - 0.105013i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.120 + 1.72i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (1.19 - 2.07i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.279 - 0.483i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.31 - 2.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.00438 + 0.00759i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
19 \( 1 - 7.57T + 19T^{2} \)
23 \( 1 + (2.51 - 4.35i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (-1.96 + 3.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.34T + 37T^{2} \)
41 \( 1 + (2.18 - 3.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.590 - 1.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.87 - 4.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.30T + 53T^{2} \)
59 \( 1 + (2.68 - 4.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.96 + 8.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.29 - 7.44i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.57T + 71T^{2} \)
73 \( 1 - 8.37T + 73T^{2} \)
79 \( 1 + (0.000301 + 0.000521i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.29 - 5.71i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.56T + 89T^{2} \)
97 \( 1 + (-7.86 - 13.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.08016870802373353880348756030, −9.985130631301947329971220932778, −9.180987326794749099127590118641, −7.75238684594892101407958972255, −7.58687196779294280537635372744, −6.49237976334345243399623489011, −5.28298615135064996561654855573, −3.63828177456579344793555930155, −2.72015519069865627170544059124, −1.32687219434723034247831057720, 0.796349043390827193154807133538, 3.30856205617723566482369333655, 4.39361018867496840777466438758, 5.23520129067738170021946854866, 6.10965313284107379681184165071, 7.53042543040196954348765975752, 8.374129863637773035272932729127, 9.021510280245520362969901282746, 9.860503662294410094768928348913, 10.69394390049583257211838932792

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