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Properties

Label 2-825-5.4-c3-0-26
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  − 5.35i·2-s − 3i·3-s − 20.6·4-s − 16.0·6-s − 19.1i·7-s + 67.8i·8-s − 9·9-s + 11·11-s + 62.0i·12-s + 37.4i·13-s − 102.·14-s + 197.·16-s + 25.2i·17-s + 48.1i·18-s + 159.·19-s + ⋯
L(s)  = 1  − 1.89i·2-s − 0.577i·3-s − 2.58·4-s − 1.09·6-s − 1.03i·7-s + 2.99i·8-s − 0.333·9-s + 0.301·11-s + 1.49i·12-s + 0.798i·13-s − 1.95·14-s + 3.09·16-s + 0.360i·17-s + 0.631i·18-s + 1.92·19-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

\(L(2)\) \(\approx\) \(1.478681659\)
\(L(\frac12)\) \(\approx\) \(1.478681659\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.35iT - 8T^{2} \)
7 \( 1 + 19.1iT - 343T^{2} \)
13 \( 1 - 37.4iT - 2.19e3T^{2} \)
17 \( 1 - 25.2iT - 4.91e3T^{2} \)
19 \( 1 - 159.T + 6.85e3T^{2} \)
23 \( 1 - 175. iT - 1.21e4T^{2} \)
29 \( 1 + 65.4T + 2.43e4T^{2} \)
31 \( 1 - 75.4T + 2.97e4T^{2} \)
37 \( 1 - 166. iT - 5.06e4T^{2} \)
41 \( 1 - 391.T + 6.89e4T^{2} \)
43 \( 1 - 63.7iT - 7.95e4T^{2} \)
47 \( 1 - 385. iT - 1.03e5T^{2} \)
53 \( 1 - 89.7iT - 1.48e5T^{2} \)
59 \( 1 + 281.T + 2.05e5T^{2} \)
61 \( 1 + 754.T + 2.26e5T^{2} \)
67 \( 1 - 168. iT - 3.00e5T^{2} \)
71 \( 1 - 950.T + 3.57e5T^{2} \)
73 \( 1 - 504. iT - 3.89e5T^{2} \)
79 \( 1 + 1.12e3T + 4.93e5T^{2} \)
83 \( 1 + 746. iT - 5.71e5T^{2} \)
89 \( 1 - 457.T + 7.04e5T^{2} \)
97 \( 1 + 1.33e3iT - 9.12e5T^{2} \)
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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

−9.579638954415154143048675525654, −9.280553634043928957963405207261, −7.965505367942767283072014519748, −7.24953182884571165318028060757, −5.79480830123398030752793004372, −4.63281469500707814845080818475, −3.75522971582822608530080405200, −2.92321873877968889744229476404, −1.53451628920443254621639679251, −0.999635919620517361532797084545, 0.52265031199242295407682000556, 2.94173697628743702137723110906, 4.18433656876199798913766120078, 5.22741138440682540785440088672, 5.63549528961169604458679627379, 6.56908704014453563452768175334, 7.57072552981682925422940797419, 8.278030578500578418999864904059, 9.154611342628056031787355377961, 9.537044758896732943455717007841

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