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Properties

Label 2-825-5.4-c3-0-23
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  + 4.91i·2-s − 3i·3-s − 16.1·4-s + 14.7·6-s + 35.2i·7-s − 39.9i·8-s − 9·9-s + 11·11-s + 48.3i·12-s + 26.9i·13-s − 173.·14-s + 67.1·16-s + 125. i·17-s − 44.2i·18-s + 134.·19-s + ⋯
L(s)  = 1  + 1.73i·2-s − 0.577i·3-s − 2.01·4-s + 1.00·6-s + 1.90i·7-s − 1.76i·8-s − 0.333·9-s + 0.301·11-s + 1.16i·12-s + 0.574i·13-s − 3.30·14-s + 1.04·16-s + 1.78i·17-s − 0.578i·18-s + 1.62·19-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

\(L(2)\) \(\approx\) \(1.541721189\)
\(L(\frac12)\) \(\approx\) \(1.541721189\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.91iT - 8T^{2} \)
7 \( 1 - 35.2iT - 343T^{2} \)
13 \( 1 - 26.9iT - 2.19e3T^{2} \)
17 \( 1 - 125. iT - 4.91e3T^{2} \)
19 \( 1 - 134.T + 6.85e3T^{2} \)
23 \( 1 - 79.5iT - 1.21e4T^{2} \)
29 \( 1 - 259.T + 2.43e4T^{2} \)
31 \( 1 - 177.T + 2.97e4T^{2} \)
37 \( 1 + 32.7iT - 5.06e4T^{2} \)
41 \( 1 + 329.T + 6.89e4T^{2} \)
43 \( 1 - 134. iT - 7.95e4T^{2} \)
47 \( 1 - 419. iT - 1.03e5T^{2} \)
53 \( 1 + 483. iT - 1.48e5T^{2} \)
59 \( 1 - 136.T + 2.05e5T^{2} \)
61 \( 1 + 623.T + 2.26e5T^{2} \)
67 \( 1 - 541. iT - 3.00e5T^{2} \)
71 \( 1 + 823.T + 3.57e5T^{2} \)
73 \( 1 - 29.0iT - 3.89e5T^{2} \)
79 \( 1 + 124.T + 4.93e5T^{2} \)
83 \( 1 + 435. iT - 5.71e5T^{2} \)
89 \( 1 - 281.T + 7.04e5T^{2} \)
97 \( 1 + 40.7iT - 9.12e5T^{2} \)
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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

−9.928202584721136240852569145439, −9.086466373829971793369431184776, −8.455056503842090727495837481585, −7.912967935137427128659862850986, −6.78020619628316735412989765546, −6.11989039456508020362318039545, −5.57161584356705175012229745609, −4.61550187881703497674887732496, −3.05996392780641534013783146702, −1.55963919262970270170784017725, 0.50139411706385640992241903535, 1.09015359937711214987059310070, 2.86004066191563273688639479159, 3.44607890462532769809026545364, 4.50704456135468620149704497625, 4.99061579993341927888377230641, 6.77551196926328372779077890855, 7.71358070605140598396208045653, 8.847988459970510667032468201967, 9.811641221807871629848785508510

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