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Properties

Label 2-825-5.4-c3-0-75
Degree $2$
Conductor $825$
Sign $0.447 + 0.894i$
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  + 2.51i·2-s + 3i·3-s + 1.66·4-s − 7.55·6-s + 18.4i·7-s + 24.3i·8-s − 9·9-s − 11·11-s + 4.98i·12-s − 39.3i·13-s − 46.4·14-s − 47.9·16-s − 125. i·17-s − 22.6i·18-s − 61.2·19-s + ⋯
L(s)  = 1  + 0.890i·2-s + 0.577i·3-s + 0.207·4-s − 0.513·6-s + 0.996i·7-s + 1.07i·8-s − 0.333·9-s − 0.301·11-s + 0.119i·12-s − 0.839i·13-s − 0.887·14-s − 0.749·16-s − 1.79i·17-s − 0.296i·18-s − 0.738·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.447 + 0.894i$
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: \((2,\ 825,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.08240232899\)
\(L(\frac12)\) \(\approx\) \(0.08240232899\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.51iT - 8T^{2} \)
7 \( 1 - 18.4iT - 343T^{2} \)
13 \( 1 + 39.3iT - 2.19e3T^{2} \)
17 \( 1 + 125. iT - 4.91e3T^{2} \)
19 \( 1 + 61.2T + 6.85e3T^{2} \)
23 \( 1 + 68.6iT - 1.21e4T^{2} \)
29 \( 1 + 201.T + 2.43e4T^{2} \)
31 \( 1 + 313.T + 2.97e4T^{2} \)
37 \( 1 + 265. iT - 5.06e4T^{2} \)
41 \( 1 - 174.T + 6.89e4T^{2} \)
43 \( 1 - 9.21iT - 7.95e4T^{2} \)
47 \( 1 - 397. iT - 1.03e5T^{2} \)
53 \( 1 - 683. iT - 1.48e5T^{2} \)
59 \( 1 - 435.T + 2.05e5T^{2} \)
61 \( 1 + 872.T + 2.26e5T^{2} \)
67 \( 1 + 594. iT - 3.00e5T^{2} \)
71 \( 1 + 921.T + 3.57e5T^{2} \)
73 \( 1 + 5.07iT - 3.89e5T^{2} \)
79 \( 1 - 1.01e3T + 4.93e5T^{2} \)
83 \( 1 + 673. iT - 5.71e5T^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + 1.50e3iT - 9.12e5T^{2} \)
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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

−9.296566392725727038053117671429, −8.952117946618407440851806125530, −7.79299789500066798971902438945, −7.23607231067152133653270214181, −5.91342828457328419971963796700, −5.55196156727081000382652128068, −4.58597673227124425313118346238, −3.03698479981035601459734017717, −2.22478679967877000496788267217, −0.01942775013708848692329849810, 1.44837573893131179794459078297, 2.05845547569887809328524444423, 3.55107777656651373261635722324, 4.11652275832888688911133599444, 5.70418405521328323967089111376, 6.70165884194313411552989904459, 7.29489096633132930478348498052, 8.257564717149259342541977899802, 9.328053427033247981666221727944, 10.28065616594497976541014187158

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