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Properties

Label 2-825-5.4-c3-0-62
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  + 1.56i·2-s − 3i·3-s + 5.56·4-s + 4.68·6-s − 10.2i·7-s + 21.1i·8-s − 9·9-s − 11·11-s − 16.6i·12-s + 40.8i·13-s + 16·14-s + 11.4·16-s − 98.7i·17-s − 14.0i·18-s + 39.6·19-s + ⋯
L(s)  = 1  + 0.552i·2-s − 0.577i·3-s + 0.695·4-s + 0.318·6-s − 0.553i·7-s + 0.935i·8-s − 0.333·9-s − 0.301·11-s − 0.401i·12-s + 0.872i·13-s + 0.305·14-s + 0.178·16-s − 1.40i·17-s − 0.184i·18-s + 0.478·19-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

\(L(2)\) \(\approx\) \(2.466827841\)
\(L(\frac12)\) \(\approx\) \(2.466827841\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.56iT - 8T^{2} \)
7 \( 1 + 10.2iT - 343T^{2} \)
13 \( 1 - 40.8iT - 2.19e3T^{2} \)
17 \( 1 + 98.7iT - 4.91e3T^{2} \)
19 \( 1 - 39.6T + 6.85e3T^{2} \)
23 \( 1 + 61.6iT - 1.21e4T^{2} \)
29 \( 1 - 149.T + 2.43e4T^{2} \)
31 \( 1 - 54.7T + 2.97e4T^{2} \)
37 \( 1 - 44.8iT - 5.06e4T^{2} \)
41 \( 1 - 336.T + 6.89e4T^{2} \)
43 \( 1 - 2.36iT - 7.95e4T^{2} \)
47 \( 1 + 333. iT - 1.03e5T^{2} \)
53 \( 1 + 640. iT - 1.48e5T^{2} \)
59 \( 1 - 370.T + 2.05e5T^{2} \)
61 \( 1 + 714.T + 2.26e5T^{2} \)
67 \( 1 + 404. iT - 3.00e5T^{2} \)
71 \( 1 - 939.T + 3.57e5T^{2} \)
73 \( 1 - 362. iT - 3.89e5T^{2} \)
79 \( 1 + 951.T + 4.93e5T^{2} \)
83 \( 1 + 735. iT - 5.71e5T^{2} \)
89 \( 1 + 385.T + 7.04e5T^{2} \)
97 \( 1 + 966. iT - 9.12e5T^{2} \)
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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

−9.756634208335109515918847414332, −8.689433553891857824650466115657, −7.81741707568331076485590608876, −7.06657404974051795656999885010, −6.61170701674766123987470769893, −5.53424029963534709263242785939, −4.55848591655620503068830589161, −3.04546132905551729018468224439, −2.08914800236058875932364256952, −0.72278701715807835849541370458, 1.09779961072852877610754903335, 2.45178039961771338546466069177, 3.23792624014631241278749856679, 4.29112399073061607463657642798, 5.60775682028971814147111514000, 6.16963853289000229626982528889, 7.44905908278291343809159284209, 8.240813991951058354199027709656, 9.257705698762947169675105758507, 10.13915588432836945707031121517

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