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Properties

Label 2-825-55.7-c1-0-2
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: $\chi_{825} (7, \cdot )$
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

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

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