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Properties

Label 2-825-5.4-c3-0-39
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  + 4i·2-s + 3i·3-s − 8·4-s − 12·6-s − 21i·7-s − 9·9-s + 11·11-s − 24i·12-s − 68i·13-s + 84·14-s − 64·16-s − 21i·17-s − 36i·18-s − 125·19-s + 63·21-s + 44i·22-s + ⋯
L(s)  = 1  + 1.41i·2-s + 0.577i·3-s − 4-s − 0.816·6-s − 1.13i·7-s − 0.333·9-s + 0.301·11-s − 0.577i·12-s − 1.45i·13-s + 1.60·14-s − 16-s − 0.299i·17-s − 0.471i·18-s − 1.50·19-s + 0.654·21-s + 0.426i·22-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

\(L(2)\) \(\approx\) \(1.860115979\)
\(L(\frac12)\) \(\approx\) \(1.860115979\)
\(L(\frac{5}{2})\) 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 \( 1 - 4iT - 8T^{2} \)
7 \( 1 + 21iT - 343T^{2} \)
13 \( 1 + 68iT - 2.19e3T^{2} \)
17 \( 1 + 21iT - 4.91e3T^{2} \)
19 \( 1 + 125T + 6.85e3T^{2} \)
23 \( 1 - 137iT - 1.21e4T^{2} \)
29 \( 1 - 150T + 2.43e4T^{2} \)
31 \( 1 - 292T + 2.97e4T^{2} \)
37 \( 1 - 349iT - 5.06e4T^{2} \)
41 \( 1 - 497T + 6.89e4T^{2} \)
43 \( 1 + 208iT - 7.95e4T^{2} \)
47 \( 1 - 369iT - 1.03e5T^{2} \)
53 \( 1 - 542iT - 1.48e5T^{2} \)
59 \( 1 + 235T + 2.05e5T^{2} \)
61 \( 1 - 482T + 2.26e5T^{2} \)
67 \( 1 - 734iT - 3.00e5T^{2} \)
71 \( 1 - 587T + 3.57e5T^{2} \)
73 \( 1 + 518iT - 3.89e5T^{2} \)
79 \( 1 - 1.04e3T + 4.93e5T^{2} \)
83 \( 1 + 608iT - 5.71e5T^{2} \)
89 \( 1 - 770T + 7.04e5T^{2} \)
97 \( 1 + 1.54e3iT - 9.12e5T^{2} \)
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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

−10.10175704855891524106993499010, −9.040568237592484271264621987660, −8.117670677337713457120576166699, −7.61922300554041846726967311750, −6.58758795558440740322585414400, −5.91318184475124063355783232224, −4.83903957345246434624335780789, −4.15074063102958659872289194961, −2.83738008635176249434543481530, −0.77658923422441761457407763300, 0.73215969055530168466454562572, 2.18811264767056741098704001441, 2.36819922164309547252848524265, 3.89257729032422102491357137513, 4.73248956341813860261875205944, 6.29370133277116369067618480684, 6.66360996620280198223327980651, 8.261142196776996641071910444847, 8.887384871684806299025557208748, 9.598312528393085659961943052962

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