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Properties

Label 2-825-11.3-c1-0-0
Degree $2$
Conductor $825$
Sign $-0.995 - 0.0981i$
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  + (0.233 − 0.169i)2-s + (−0.309 − 0.951i)3-s + (−0.592 + 1.82i)4-s + (−0.233 − 0.169i)6-s + (−1.13 + 3.48i)7-s + (0.349 + 1.07i)8-s + (−0.809 + 0.587i)9-s + (1.25 − 3.06i)11-s + 1.91·12-s + (−3.90 + 2.83i)13-s + (0.327 + 1.00i)14-s + (−2.83 − 2.06i)16-s + (−3.10 − 2.25i)17-s + (−0.0892 + 0.274i)18-s + (−2.15 − 6.62i)19-s + ⋯
L(s)  = 1  + (0.165 − 0.120i)2-s + (−0.178 − 0.549i)3-s + (−0.296 + 0.911i)4-s + (−0.0953 − 0.0693i)6-s + (−0.428 + 1.31i)7-s + (0.123 + 0.380i)8-s + (−0.269 + 0.195i)9-s + (0.378 − 0.925i)11-s + 0.553·12-s + (−1.08 + 0.786i)13-s + (0.0874 + 0.269i)14-s + (−0.709 − 0.515i)16-s + (−0.753 − 0.547i)17-s + (−0.0210 + 0.0647i)18-s + (−0.493 − 1.51i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.995 - 0.0981i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.995 - 0.0981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0113893 + 0.231605i\)
\(L(\frac12)\) \(\approx\) \(0.0113893 + 0.231605i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 \)
11 \( 1 + (-1.25 + 3.06i)T \)
good2 \( 1 + (-0.233 + 0.169i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (1.13 - 3.48i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.90 - 2.83i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.10 + 2.25i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.15 + 6.62i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.20T + 23T^{2} \)
29 \( 1 + (-1.52 + 4.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.96 - 4.33i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.737 - 2.27i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.33 - 4.09i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.772T + 43T^{2} \)
47 \( 1 + (1.99 + 6.15i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.70 - 6.32i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.48 - 13.7i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.48 - 1.07i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 6.10T + 67T^{2} \)
71 \( 1 + (-2.66 - 1.93i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.0591 + 0.182i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.55 + 1.85i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.23 + 0.899i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 3.04T + 89T^{2} \)
97 \( 1 + (11.1 - 8.09i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.97371480709721389915022998264, −9.326653111942389621026336545145, −9.034009523661137372293590772486, −8.200444260038062409865197603106, −7.09018662725538701675288444964, −6.44040595303388993031286770523, −5.28797930477533160309916033965, −4.37417078735780767242237465632, −2.94038828157338976271828963923, −2.31268009611474887421025347451, 0.10410858524954534746344925264, 1.80883513954485136782739284299, 3.69395776835535172444229899722, 4.36940007657081957799381064236, 5.25746556447180843941516802637, 6.28650415761230663590360522336, 7.06711065791845673602330549793, 8.041803408770021415616501491286, 9.402783450829838684417777615151, 9.879206988297729080560525150276

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