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Properties

Label 2-825-11.5-c1-0-21
Degree $2$
Conductor $825$
Sign $0.501 + 0.865i$
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  + (−0.338 − 1.04i)2-s + (0.809 + 0.587i)3-s + (0.647 − 0.470i)4-s + (0.338 − 1.04i)6-s + (−0.570 + 0.414i)7-s + (−2.48 − 1.80i)8-s + (0.309 + 0.951i)9-s + (3.31 + 0.189i)11-s + 0.800·12-s + (1.45 + 4.48i)13-s + (0.624 + 0.453i)14-s + (−0.543 + 1.67i)16-s + (2.40 − 7.39i)17-s + (0.886 − 0.643i)18-s + (0.970 + 0.705i)19-s + ⋯
L(s)  = 1  + (−0.239 − 0.736i)2-s + (0.467 + 0.339i)3-s + (0.323 − 0.235i)4-s + (0.138 − 0.425i)6-s + (−0.215 + 0.156i)7-s + (−0.877 − 0.637i)8-s + (0.103 + 0.317i)9-s + (0.998 + 0.0572i)11-s + 0.231·12-s + (0.403 + 1.24i)13-s + (0.166 + 0.121i)14-s + (−0.135 + 0.418i)16-s + (0.583 − 1.79i)17-s + (0.208 − 0.151i)18-s + (0.222 + 0.161i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.501 + 0.865i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.501 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62987 - 0.939190i\)
\(L(\frac12)\) \(\approx\) \(1.62987 - 0.939190i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 \)
11 \( 1 + (-3.31 - 0.189i)T \)
good2 \( 1 + (0.338 + 1.04i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (0.570 - 0.414i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.45 - 4.48i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.40 + 7.39i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.970 - 0.705i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 6.89T + 23T^{2} \)
29 \( 1 + (1.07 - 0.780i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.37 + 7.30i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.82 + 4.95i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.188 - 0.136i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.32T + 43T^{2} \)
47 \( 1 + (-6.73 - 4.89i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.10 + 6.48i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.86 + 2.08i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.35 - 10.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.04T + 67T^{2} \)
71 \( 1 + (-0.207 + 0.637i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.04 - 2.94i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.704 - 2.16i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.652 + 2.00i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 3.34T + 89T^{2} \)
97 \( 1 + (1.02 + 3.15i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.837987678349685618954745592184, −9.373582383685058327703328226806, −8.930632260636176241793833957648, −7.42284724565012155536823619775, −6.73234227764721756372746397712, −5.69356160909298319576976658914, −4.41020386894498384799147249883, −3.37417983498713909577887758099, −2.43742444482347952221271424747, −1.15711050317618743105927500844, 1.38275823760638317943233582535, 3.00440169890153358132491893410, 3.70389639723196781658080592606, 5.36920032562334589095387526369, 6.30098482124828726054646131298, 6.91385608646615906612101798842, 7.87962477979934121270594690413, 8.452052576698726923447508937240, 9.186351030663079644078249929857, 10.32503473819878474991733241281

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