# Properties

 Label 2-825-1.1-c5-0-86 Degree $2$ Conductor $825$ Sign $-1$ Analytic cond. $132.316$ Root an. cond. $11.5028$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 10.0·2-s − 9·3-s + 68.9·4-s + 90.4·6-s + 29.0·7-s − 370.·8-s + 81·9-s + 121·11-s − 620.·12-s − 1.02e3·13-s − 291.·14-s + 1.52e3·16-s + 1.50e3·17-s − 813.·18-s + 1.64e3·19-s − 261.·21-s − 1.21e3·22-s − 1.47e3·23-s + 3.33e3·24-s + 1.02e4·26-s − 729·27-s + 2.00e3·28-s − 4.57e3·29-s + 7.53e3·31-s − 3.40e3·32-s − 1.08e3·33-s − 1.51e4·34-s + ⋯
 L(s)  = 1 − 1.77·2-s − 0.577·3-s + 2.15·4-s + 1.02·6-s + 0.224·7-s − 2.04·8-s + 0.333·9-s + 0.301·11-s − 1.24·12-s − 1.67·13-s − 0.397·14-s + 1.48·16-s + 1.26·17-s − 0.591·18-s + 1.04·19-s − 0.129·21-s − 0.535·22-s − 0.582·23-s + 1.18·24-s + 2.98·26-s − 0.192·27-s + 0.482·28-s − 1.00·29-s + 1.40·31-s − 0.588·32-s − 0.174·33-s − 2.25·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$825$$    =    $$3 \cdot 5^{2} \cdot 11$$ Sign: $-1$ Analytic conductor: $$132.316$$ Root analytic conductor: $$11.5028$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 825,\ (\ :5/2),\ -1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 9T$$
5 $$1$$
11 $$1 - 121T$$
good2 $$1 + 10.0T + 32T^{2}$$
7 $$1 - 29.0T + 1.68e4T^{2}$$
13 $$1 + 1.02e3T + 3.71e5T^{2}$$
17 $$1 - 1.50e3T + 1.41e6T^{2}$$
19 $$1 - 1.64e3T + 2.47e6T^{2}$$
23 $$1 + 1.47e3T + 6.43e6T^{2}$$
29 $$1 + 4.57e3T + 2.05e7T^{2}$$
31 $$1 - 7.53e3T + 2.86e7T^{2}$$
37 $$1 + 4.40e3T + 6.93e7T^{2}$$
41 $$1 - 5.62e3T + 1.15e8T^{2}$$
43 $$1 - 2.28e3T + 1.47e8T^{2}$$
47 $$1 + 5.98e3T + 2.29e8T^{2}$$
53 $$1 + 2.84e4T + 4.18e8T^{2}$$
59 $$1 - 2.59e4T + 7.14e8T^{2}$$
61 $$1 + 5.13e4T + 8.44e8T^{2}$$
67 $$1 - 3.91e4T + 1.35e9T^{2}$$
71 $$1 - 3.75e4T + 1.80e9T^{2}$$
73 $$1 + 5.86e4T + 2.07e9T^{2}$$
79 $$1 + 8.27e3T + 3.07e9T^{2}$$
83 $$1 - 2.13e4T + 3.93e9T^{2}$$
89 $$1 - 7.83e4T + 5.58e9T^{2}$$
97 $$1 + 6.45e4T + 8.58e9T^{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

−9.368062234102189195438552596940, −8.050742227031890438222820570905, −7.60735179604719338816571931994, −6.82436348068163103312596629269, −5.78965211775070673157921535766, −4.76957189283136033481948035707, −3.12245203968317520079696524477, −1.93389099870098101383101916392, −0.970374087862571651213944914098, 0, 0.970374087862571651213944914098, 1.93389099870098101383101916392, 3.12245203968317520079696524477, 4.76957189283136033481948035707, 5.78965211775070673157921535766, 6.82436348068163103312596629269, 7.60735179604719338816571931994, 8.050742227031890438222820570905, 9.368062234102189195438552596940