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Properties

Label 2-825-5.4-c3-0-30
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  + 0.540i·2-s + 3i·3-s + 7.70·4-s − 1.62·6-s + 24.4i·7-s + 8.49i·8-s − 9·9-s − 11·11-s + 23.1i·12-s + 84.5i·13-s − 13.2·14-s + 57.0·16-s − 62.8i·17-s − 4.86i·18-s + 159.·19-s + ⋯
L(s)  = 1  + 0.191i·2-s + 0.577i·3-s + 0.963·4-s − 0.110·6-s + 1.32i·7-s + 0.375i·8-s − 0.333·9-s − 0.301·11-s + 0.556i·12-s + 1.80i·13-s − 0.252·14-s + 0.891·16-s − 0.895i·17-s − 0.0637i·18-s + 1.92·19-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

\(L(2)\) \(\approx\) \(2.321316530\)
\(L(\frac12)\) \(\approx\) \(2.321316530\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.540iT - 8T^{2} \)
7 \( 1 - 24.4iT - 343T^{2} \)
13 \( 1 - 84.5iT - 2.19e3T^{2} \)
17 \( 1 + 62.8iT - 4.91e3T^{2} \)
19 \( 1 - 159.T + 6.85e3T^{2} \)
23 \( 1 - 114. iT - 1.21e4T^{2} \)
29 \( 1 + 172.T + 2.43e4T^{2} \)
31 \( 1 - 8.87T + 2.97e4T^{2} \)
37 \( 1 + 14.9iT - 5.06e4T^{2} \)
41 \( 1 + 463.T + 6.89e4T^{2} \)
43 \( 1 + 486. iT - 7.95e4T^{2} \)
47 \( 1 + 118. iT - 1.03e5T^{2} \)
53 \( 1 - 273. iT - 1.48e5T^{2} \)
59 \( 1 - 884.T + 2.05e5T^{2} \)
61 \( 1 + 347.T + 2.26e5T^{2} \)
67 \( 1 - 720. iT - 3.00e5T^{2} \)
71 \( 1 + 71.7T + 3.57e5T^{2} \)
73 \( 1 + 146. iT - 3.89e5T^{2} \)
79 \( 1 + 147.T + 4.93e5T^{2} \)
83 \( 1 - 399. iT - 5.71e5T^{2} \)
89 \( 1 + 1.02e3T + 7.04e5T^{2} \)
97 \( 1 - 1.64e3iT - 9.12e5T^{2} \)
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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

−10.03266974852050377146506018917, −9.346401058415568863026100844531, −8.682343840892149095880852722874, −7.45187861585602595325932160777, −6.83497997958960063962460512964, −5.56681186888794644072019802818, −5.25311387702678813466624395019, −3.67848588165221376444947054006, −2.65784172380986433086446510095, −1.70299599475119007294995153869, 0.58343602511185087734413377650, 1.48015302299639405485927326152, 2.91107282603252734793412222299, 3.60453612947289667477787866720, 5.17084284230164884074195361234, 6.08633313699047049618042836796, 7.04273455640921417268723015246, 7.68556880706194704680384664197, 8.218335201090754505809693856298, 9.867869191335433250829777563873

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