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Properties

Label 2-825-11.9-c1-0-13
Degree $2$
Conductor $825$
Sign $-0.0536 - 0.998i$
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  + (−0.150 + 0.464i)2-s + (−0.809 + 0.587i)3-s + (1.42 + 1.03i)4-s + (−0.150 − 0.464i)6-s + (4.13 + 3.00i)7-s + (−1.48 + 1.07i)8-s + (0.309 − 0.951i)9-s + (2.66 − 1.97i)11-s − 1.76·12-s + (−0.561 + 1.72i)13-s + (−2.01 + 1.46i)14-s + (0.811 + 2.49i)16-s + (−0.197 − 0.608i)17-s + (0.395 + 0.287i)18-s + (3.55 − 2.58i)19-s + ⋯
L(s)  = 1  + (−0.106 + 0.328i)2-s + (−0.467 + 0.339i)3-s + (0.712 + 0.517i)4-s + (−0.0616 − 0.189i)6-s + (1.56 + 1.13i)7-s + (−0.525 + 0.381i)8-s + (0.103 − 0.317i)9-s + (0.804 − 0.594i)11-s − 0.508·12-s + (−0.155 + 0.479i)13-s + (−0.539 + 0.391i)14-s + (0.202 + 0.624i)16-s + (−0.0479 − 0.147i)17-s + (0.0931 + 0.0676i)18-s + (0.816 − 0.593i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0536 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0536 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.0536 - 0.998i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (526, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.0536 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25447 + 1.32370i\)
\(L(\frac12)\) \(\approx\) \(1.25447 + 1.32370i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.66 + 1.97i)T \)
good2 \( 1 + (0.150 - 0.464i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-4.13 - 3.00i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.561 - 1.72i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.197 + 0.608i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.55 + 2.58i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.37T + 23T^{2} \)
29 \( 1 + (6.11 + 4.44i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.681 + 2.09i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.83 + 3.51i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.80 + 1.31i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.38T + 43T^{2} \)
47 \( 1 + (-1.22 + 0.890i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.83 - 8.72i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (4.34 + 3.15i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.24 - 6.91i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 7.25T + 67T^{2} \)
71 \( 1 + (-1.70 - 5.23i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.77 + 4.19i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.588 + 1.81i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.0500 + 0.153i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 1.19T + 89T^{2} \)
97 \( 1 + (-4.41 + 13.5i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.79274333175780882050356594376, −9.233171684256154354338476381183, −8.843137491251926819155078801499, −7.84297993302844910360608796667, −7.06062347299629912122256315920, −5.96597736331411386995078717351, −5.32532189420997490060522123981, −4.23718079874961320851587882131, −2.87335349036795187618588466700, −1.66803558564197038692565353795, 1.15390620009759827987851334462, 1.77114962062176195418986647569, 3.48096610831853298440246496289, 4.77079027240403550377793086299, 5.48415791654095850597964164783, 6.76686077551023233781527911267, 7.28586556260057365985217744989, 8.093554419900376112514991683601, 9.375021931596445948551791851128, 10.34500874634034647964061340751

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