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Properties

Label 2-825-11.4-c1-0-16
Degree $2$
Conductor $825$
Sign $-0.966 - 0.255i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.24 + 1.62i)2-s + (−0.309 + 0.951i)3-s + (1.75 + 5.40i)4-s + (−2.24 + 1.62i)6-s + (0.703 + 2.16i)7-s + (−3.15 + 9.71i)8-s + (−0.809 − 0.587i)9-s + (−0.105 − 3.31i)11-s − 5.68·12-s + (−0.352 − 0.256i)13-s + (−1.95 + 6.00i)14-s + (−13.7 + 9.96i)16-s + (4.04 − 2.93i)17-s + (−0.856 − 2.63i)18-s + (1.45 − 4.46i)19-s + ⋯
L(s)  = 1  + (1.58 + 1.15i)2-s + (−0.178 + 0.549i)3-s + (0.878 + 2.70i)4-s + (−0.915 + 0.665i)6-s + (0.266 + 0.818i)7-s + (−1.11 + 3.43i)8-s + (−0.269 − 0.195i)9-s + (−0.0317 − 0.999i)11-s − 1.64·12-s + (−0.0977 − 0.0710i)13-s + (−0.521 + 1.60i)14-s + (−3.42 + 2.49i)16-s + (0.981 − 0.712i)17-s + (−0.201 − 0.621i)18-s + (0.332 − 1.02i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.966 - 0.255i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.966 - 0.255i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.454837 + 3.49724i\)
\(L(\frac12)\) \(\approx\) \(0.454837 + 3.49724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 \)
11 \( 1 + (0.105 + 3.31i)T \)
good2 \( 1 + (-2.24 - 1.62i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (-0.703 - 2.16i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.352 + 0.256i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.04 + 2.93i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.45 + 4.46i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.845T + 23T^{2} \)
29 \( 1 + (0.821 + 2.52i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.77 - 2.74i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.73 - 8.42i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.32 - 4.08i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.00T + 43T^{2} \)
47 \( 1 + (0.144 - 0.445i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (8.76 + 6.37i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.21 - 3.74i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.39 - 1.74i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 2.47T + 67T^{2} \)
71 \( 1 + (-9.15 + 6.65i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.60 + 8.01i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.79 - 6.38i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.78 - 3.47i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 5.89T + 89T^{2} \)
97 \( 1 + (6.99 + 5.08i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.07486922485139381097811416500, −9.581772447163898172125197710297, −8.521954772274143716006061132572, −7.961670127673698034376231375547, −6.81003771687662692132913602305, −6.03921183588100069740940177808, −5.20021916392517406043983060158, −4.75437235309880314084245417525, −3.39263954382239583992676463162, −2.78628428175166812179378958449, 1.18616893066572821877535260385, 2.10220126901709932379634924783, 3.43813190745573116827641858160, 4.24219274713163718573979944734, 5.18954517220457016304041743955, 6.01275652638458296395046560158, 6.96829543205724313296944129423, 7.81970572335325383446375763644, 9.535987226311662847464090498601, 10.24964141404726256947935600816

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