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Properties

Label 2-528-44.43-c5-0-26
Degree $2$
Conductor $528$
Sign $0.996 + 0.0812i$
Analytic cond. $84.6826$
Root an. cond. $9.20231$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9i·3-s − 84.4·5-s + 98.5·7-s − 81·9-s + (−362. + 171. i)11-s − 28.0i·13-s − 760. i·15-s + 334. i·17-s − 2.60e3·19-s + 887. i·21-s − 1.42e3i·23-s + 4.01e3·25-s − 729i·27-s + 2.77e3i·29-s − 4.56e3i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.51·5-s + 0.760·7-s − 0.333·9-s + (−0.903 + 0.427i)11-s − 0.0460i·13-s − 0.872i·15-s + 0.280i·17-s − 1.65·19-s + 0.439i·21-s − 0.563i·23-s + 1.28·25-s − 0.192i·27-s + 0.611i·29-s − 0.853i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0812i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
Sign: $0.996 + 0.0812i$
Analytic conductor: \(84.6826\)
Root analytic conductor: \(9.20231\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{528} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 528,\ (\ :5/2),\ 0.996 + 0.0812i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8112323997\)
\(L(\frac12)\) \(\approx\) \(0.8112323997\)
\(L(\frac{7}{2})\) 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 \( 1 - 9iT \)
11 \( 1 + (362. - 171. i)T \)
good5 \( 1 + 84.4T + 3.12e3T^{2} \)
7 \( 1 - 98.5T + 1.68e4T^{2} \)
13 \( 1 + 28.0iT - 3.71e5T^{2} \)
17 \( 1 - 334. iT - 1.41e6T^{2} \)
19 \( 1 + 2.60e3T + 2.47e6T^{2} \)
23 \( 1 + 1.42e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.77e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.56e3iT - 2.86e7T^{2} \)
37 \( 1 + 5.08e3T + 6.93e7T^{2} \)
41 \( 1 - 4.62e3iT - 1.15e8T^{2} \)
43 \( 1 + 8.78e3T + 1.47e8T^{2} \)
47 \( 1 - 621. iT - 2.29e8T^{2} \)
53 \( 1 - 8.70e3T + 4.18e8T^{2} \)
59 \( 1 - 4.33e4iT - 7.14e8T^{2} \)
61 \( 1 + 4.35e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.01e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.36e4iT - 1.80e9T^{2} \)
73 \( 1 + 7.79e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.75e4T + 3.07e9T^{2} \)
83 \( 1 - 8.53e4T + 3.93e9T^{2} \)
89 \( 1 - 4.83e4T + 5.58e9T^{2} \)
97 \( 1 - 1.51e5T + 8.58e9T^{2} \)
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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

−10.38543119650045634820638572170, −8.965592816357790317348223570076, −8.164921880881741064590407440471, −7.66880512618157409195207567076, −6.44943237064832536286861317865, −4.99664997560473155177671829222, −4.41531458683480692101849101034, −3.47273437655374853306447497994, −2.12917867951232127981762195893, −0.33729434212608270614960895238, 0.55582954798108947620197214524, 2.01562729754575243750662627161, 3.29683850269881306810276563839, 4.36335389848929347243035095119, 5.32076596724066622555513627553, 6.63791463151393546138178868711, 7.55869129143439112436008286183, 8.190447173191458055989314315531, 8.739796772729584112399524824685, 10.35423991300917832164389368770

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