Properties

Label 2-825-5.4-c3-0-50
Degree $2$
Conductor $825$
Sign $-0.894 + 0.447i$
Analytic cond. $48.6765$
Root an. cond. $6.97686$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.97i·2-s + 3i·3-s − 16.7·4-s + 14.9·6-s − 5.48i·7-s + 43.3i·8-s − 9·9-s + 11·11-s − 50.1i·12-s + 24.5i·13-s − 27.2·14-s + 81.6·16-s + 59.3i·17-s + 44.7i·18-s − 5.89·19-s + ⋯
L(s)  = 1  − 1.75i·2-s + 0.577i·3-s − 2.08·4-s + 1.01·6-s − 0.296i·7-s + 1.91i·8-s − 0.333·9-s + 0.301·11-s − 1.20i·12-s + 0.524i·13-s − 0.520·14-s + 1.27·16-s + 0.847i·17-s + 0.585i·18-s − 0.0711·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(48.6765\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :3/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.433340734\)
\(L(\frac12)\) \(\approx\) \(1.433340734\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3iT \)
5 \( 1 \)
11 \( 1 - 11T \)
good2 \( 1 + 4.97iT - 8T^{2} \)
7 \( 1 + 5.48iT - 343T^{2} \)
13 \( 1 - 24.5iT - 2.19e3T^{2} \)
17 \( 1 - 59.3iT - 4.91e3T^{2} \)
19 \( 1 + 5.89T + 6.85e3T^{2} \)
23 \( 1 + 68.4iT - 1.21e4T^{2} \)
29 \( 1 - 265.T + 2.43e4T^{2} \)
31 \( 1 + 196.T + 2.97e4T^{2} \)
37 \( 1 + 166. iT - 5.06e4T^{2} \)
41 \( 1 - 424.T + 6.89e4T^{2} \)
43 \( 1 + 177. iT - 7.95e4T^{2} \)
47 \( 1 + 141. iT - 1.03e5T^{2} \)
53 \( 1 + 339. iT - 1.48e5T^{2} \)
59 \( 1 + 416.T + 2.05e5T^{2} \)
61 \( 1 - 662.T + 2.26e5T^{2} \)
67 \( 1 + 313. iT - 3.00e5T^{2} \)
71 \( 1 + 153.T + 3.57e5T^{2} \)
73 \( 1 - 153. iT - 3.89e5T^{2} \)
79 \( 1 + 403.T + 4.93e5T^{2} \)
83 \( 1 + 652. iT - 5.71e5T^{2} \)
89 \( 1 + 1.22e3T + 7.04e5T^{2} \)
97 \( 1 + 959. iT - 9.12e5T^{2} \)
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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

−9.719403420406324237010382383421, −8.995741904051576072763499012853, −8.288514750313401562848550273326, −6.85178230249117176547417292095, −5.57489993554492383010913253612, −4.37429123430493752225928330783, −3.94242892453941661332561963177, −2.83073795781045484529770970591, −1.77105345417228666139525434763, −0.49894755797449731922978876321, 0.931091076678150464713588190043, 2.78316261502009388736625661354, 4.26486558659487439820287666440, 5.28277252274552073245733820714, 5.97123731429077855535952746905, 6.81363901096238439245231268137, 7.50491342153128530059579954211, 8.222442823441254634169944053694, 9.014223582511925788466147782831, 9.735129307905865494755214278691

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