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Properties

Label 2-825-165.53-c1-0-17
Degree $2$
Conductor $825$
Sign $0.0177 - 0.999i$
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  + (−2.41 + 1.22i)2-s + (−1.72 + 0.140i)3-s + (3.13 − 4.31i)4-s + (3.99 − 2.45i)6-s + (3.61 − 0.571i)7-s + (−1.40 + 8.89i)8-s + (2.96 − 0.484i)9-s + (−2.81 + 1.75i)11-s + (−4.80 + 7.87i)12-s + (−3.40 + 1.73i)13-s + (−8.00 + 5.81i)14-s + (−4.24 − 13.0i)16-s + (1.05 − 2.08i)17-s + (−6.54 + 4.80i)18-s + (1.93 + 2.67i)19-s + ⋯
L(s)  = 1  + (−1.70 + 0.868i)2-s + (−0.996 + 0.0809i)3-s + (1.56 − 2.15i)4-s + (1.62 − 1.00i)6-s + (1.36 − 0.216i)7-s + (−0.498 + 3.14i)8-s + (0.986 − 0.161i)9-s + (−0.848 + 0.529i)11-s + (−1.38 + 2.27i)12-s + (−0.943 + 0.480i)13-s + (−2.13 + 1.55i)14-s + (−1.06 − 3.26i)16-s + (0.257 − 0.504i)17-s + (−1.54 + 1.13i)18-s + (0.445 + 0.612i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0177 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0177 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.0177 - 0.999i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.0177 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.349178 + 0.343031i\)
\(L(\frac12)\) \(\approx\) \(0.349178 + 0.343031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.140i)T \)
5 \( 1 \)
11 \( 1 + (2.81 - 1.75i)T \)
good2 \( 1 + (2.41 - 1.22i)T + (1.17 - 1.61i)T^{2} \)
7 \( 1 + (-3.61 + 0.571i)T + (6.65 - 2.16i)T^{2} \)
13 \( 1 + (3.40 - 1.73i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-1.05 + 2.08i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (-1.93 - 2.67i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-3.29 + 3.29i)T - 23iT^{2} \)
29 \( 1 + (-5.78 - 4.19i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.163 - 0.503i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.47 - 0.391i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (2.40 + 3.30i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-0.617 - 0.617i)T + 43iT^{2} \)
47 \( 1 + (-6.72 - 1.06i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (0.637 + 1.25i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (7.62 + 5.53i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.64 - 8.15i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-0.630 + 0.630i)T - 67iT^{2} \)
71 \( 1 + (2.83 - 0.920i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.01 + 6.40i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (-9.47 - 3.07i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-10.1 - 5.16i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 - 0.794T + 89T^{2} \)
97 \( 1 + (-4.81 - 9.45i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.42771694042057247468378692664, −9.617421860097132151736999672820, −8.681709354150935067233765194946, −7.65442568926727515977923481402, −7.32836164220017268205956991598, −6.40624714231803579915342904740, −5.15081111571558744638921813900, −4.89331208059237601666616119848, −2.14988225385597127491245314264, −0.986649712380287355680308704479, 0.63311037357756971527238804565, 1.81857422722394911177740860624, 2.97822810364235278125880375454, 4.63884604984196063283844275917, 5.63960354274187297293754651244, 7.01802413249619560686114560007, 7.76041171823826481034019525298, 8.276409385904853659297236465063, 9.351511974685364711310180328259, 10.23315418224103944269506706653

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