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Properties

Label 2-825-55.8-c1-0-22
Degree $2$
Conductor $825$
Sign $-0.873 + 0.486i$
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  + (−2.13 − 0.338i)2-s + (0.453 − 0.891i)3-s + (2.56 + 0.832i)4-s + (−1.27 + 1.75i)6-s + (0.770 − 0.392i)7-s + (−1.33 − 0.682i)8-s + (−0.587 − 0.809i)9-s + (−2.59 + 2.06i)11-s + (1.90 − 1.90i)12-s + (0.604 − 3.81i)13-s + (−1.78 + 0.578i)14-s + (−1.72 − 1.25i)16-s + (0.752 + 4.74i)17-s + (0.983 + 1.93i)18-s + (−0.838 − 2.58i)19-s + ⋯
L(s)  = 1  + (−1.51 − 0.239i)2-s + (0.262 − 0.514i)3-s + (1.28 + 0.416i)4-s + (−0.519 + 0.715i)6-s + (0.291 − 0.148i)7-s + (−0.473 − 0.241i)8-s + (−0.195 − 0.269i)9-s + (−0.783 + 0.621i)11-s + (0.549 − 0.549i)12-s + (0.167 − 1.05i)13-s + (−0.475 + 0.154i)14-s + (−0.430 − 0.312i)16-s + (0.182 + 1.15i)17-s + (0.231 + 0.455i)18-s + (−0.192 − 0.591i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.873 + 0.486i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.873 + 0.486i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117674 - 0.453187i\)
\(L(\frac12)\) \(\approx\) \(0.117674 - 0.453187i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.453 + 0.891i)T \)
5 \( 1 \)
11 \( 1 + (2.59 - 2.06i)T \)
good2 \( 1 + (2.13 + 0.338i)T + (1.90 + 0.618i)T^{2} \)
7 \( 1 + (-0.770 + 0.392i)T + (4.11 - 5.66i)T^{2} \)
13 \( 1 + (-0.604 + 3.81i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (-0.752 - 4.74i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (0.838 + 2.58i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.79 + 4.79i)T + 23iT^{2} \)
29 \( 1 + (-1.50 + 4.61i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.834 + 0.606i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.97 + 3.87i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (-3.96 + 1.28i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-7.22 + 7.22i)T - 43iT^{2} \)
47 \( 1 + (6.70 + 3.41i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (6.46 + 1.02i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-7.48 - 2.43i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.304 + 0.418i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-4.30 + 4.30i)T - 67iT^{2} \)
71 \( 1 + (13.0 + 9.46i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.974 + 1.91i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (3.33 - 2.42i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (13.4 - 2.13i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + 4.64iT - 89T^{2} \)
97 \( 1 + (1.43 - 9.04i)T + (-92.2 - 29.9i)T^{2} \)
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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

−9.961755275459891673163487965163, −8.902909430310377785615048302090, −8.073556715913189770448696762650, −7.81800430703614572454637187821, −6.78902679497817773474004277038, −5.69747297567438445545013073858, −4.32144269490391897817803896854, −2.72289756773954118012783143299, −1.81589933758989073633130775110, −0.37003235153570671245372549822, 1.51063357484911874021099194837, 2.84884881077473581554854809127, 4.25309692628536771415645325361, 5.40792997001543177745568845869, 6.52406496498800634549948251559, 7.54905104797199128456746229416, 8.187167161172280672856344212766, 8.907643362273969581104762437210, 9.671666166400448700724495973887, 10.20339743122150665692220270298

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