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Properties

Label 2-528-33.29-c1-0-5
Degree $2$
Conductor $528$
Sign $-0.315 - 0.948i$
Analytic cond. $4.21610$
Root an. cond. $2.05331$
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  + (0.309 + 1.70i)3-s + (0.442 + 0.609i)5-s + (0.442 − 0.143i)7-s + (−2.80 + 1.05i)9-s + (2.46 + 2.21i)11-s + (−3.12 + 4.29i)13-s + (−0.901 + 0.942i)15-s + (2.99 − 2.17i)17-s + (−2.02 − 0.659i)19-s + (0.381 + 0.710i)21-s + 6.24i·23-s + (1.36 − 4.21i)25-s + (−2.66 − 4.46i)27-s + (3.09 + 9.53i)29-s + (−2.96 − 2.15i)31-s + ⋯
L(s)  = 1  + (0.178 + 0.983i)3-s + (0.197 + 0.272i)5-s + (0.167 − 0.0543i)7-s + (−0.936 + 0.351i)9-s + (0.744 + 0.668i)11-s + (−0.865 + 1.19i)13-s + (−0.232 + 0.243i)15-s + (0.726 − 0.527i)17-s + (−0.465 − 0.151i)19-s + (0.0833 + 0.154i)21-s + 1.30i·23-s + (0.273 − 0.843i)25-s + (−0.512 − 0.858i)27-s + (0.575 + 1.77i)29-s + (−0.532 − 0.386i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $-0.315 - 0.948i$
Analytic conductor: \(4.21610\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :1/2),\ -0.315 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.850430 + 1.17871i\)
\(L(\frac12)\) \(\approx\) \(0.850430 + 1.17871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.309 - 1.70i)T \)
11 \( 1 + (-2.46 - 2.21i)T \)
good5 \( 1 + (-0.442 - 0.609i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.442 + 0.143i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.12 - 4.29i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.99 + 2.17i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.02 + 0.659i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.24iT - 23T^{2} \)
29 \( 1 + (-3.09 - 9.53i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.96 + 2.15i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.16 + 6.66i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.0135 + 0.0416i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.49iT - 43T^{2} \)
47 \( 1 + (3.03 + 0.987i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.00 + 4.13i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-11.0 + 3.59i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.31 - 3.19i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.70T + 67T^{2} \)
71 \( 1 + (0.527 + 0.726i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-7.32 + 2.38i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.34 + 3.22i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.76 + 6.36i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 6.48iT - 89T^{2} \)
97 \( 1 + (-2.13 - 1.55i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.08925240566920150037804781634, −10.02394984443618977309228538086, −9.499431292760650386554435778928, −8.741246260639576540380398509646, −7.46527326057764489911837943268, −6.61178051614655248456378401230, −5.28194075190092345579561191308, −4.48161629826686457025522139752, −3.42228241832282648531488898417, −2.04738776923316972544956776289, 0.861118362337336697328408735705, 2.34288608605806259539238535444, 3.55668518736839375678354013918, 5.14052084540614745841713641452, 6.05221373707169542028935851208, 6.92136879230498974910031681701, 8.093374609698445159674943225240, 8.471208204089279386756001877136, 9.661094768923171562732979689192, 10.60720578796230748475271333942

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