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Properties

Label 2-825-11.3-c1-0-17
Degree $2$
Conductor $825$
Sign $0.656 - 0.754i$
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  + (1.50 − 1.09i)2-s + (0.309 + 0.951i)3-s + (0.453 − 1.39i)4-s + (1.50 + 1.09i)6-s + (−0.812 + 2.50i)7-s + (0.306 + 0.943i)8-s + (−0.809 + 0.587i)9-s + (1.77 + 2.80i)11-s + 1.46·12-s + (−5.24 + 3.80i)13-s + (1.51 + 4.65i)14-s + (3.86 + 2.81i)16-s + (−3.82 − 2.77i)17-s + (−0.575 + 1.77i)18-s + (0.164 + 0.506i)19-s + ⋯
L(s)  = 1  + (1.06 − 0.773i)2-s + (0.178 + 0.549i)3-s + (0.226 − 0.697i)4-s + (0.614 + 0.446i)6-s + (−0.307 + 0.945i)7-s + (0.108 + 0.333i)8-s + (−0.269 + 0.195i)9-s + (0.534 + 0.845i)11-s + 0.423·12-s + (−1.45 + 1.05i)13-s + (0.404 + 1.24i)14-s + (0.967 + 0.702i)16-s + (−0.926 − 0.673i)17-s + (−0.135 + 0.417i)18-s + (0.0377 + 0.116i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.656 - 0.754i$
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: \((2,\ 825,\ (\ :1/2),\ 0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28960 + 1.04221i\)
\(L(\frac12)\) \(\approx\) \(2.28960 + 1.04221i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.77 - 2.80i)T \)
good2 \( 1 + (-1.50 + 1.09i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.812 - 2.50i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (5.24 - 3.80i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.82 + 2.77i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.164 - 0.506i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.54T + 23T^{2} \)
29 \( 1 + (-3.28 + 10.1i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.154 - 0.112i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.30 + 4.02i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.187 + 0.575i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.24T + 43T^{2} \)
47 \( 1 + (-2.72 - 8.39i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.41 + 3.93i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.83 - 5.63i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.60 - 1.89i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.22T + 67T^{2} \)
71 \( 1 + (-2.21 - 1.61i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.84 + 14.9i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.979 - 0.711i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.77 - 2.74i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.40T + 89T^{2} \)
97 \( 1 + (5.92 - 4.30i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.48764833367691103591562201307, −9.325838054361912316859699262060, −9.217710782831004322792720249858, −7.75056945066388958945685339483, −6.68918908679024194416212936770, −5.56847871154129978267618762525, −4.58919231493932081994718865387, −4.19806566875735060191336296178, −2.66898585899728789429365750946, −2.25612105876794567275857140970, 0.872025677249746738072860978454, 2.87233964116577775348455802586, 3.80579340599987030129738980257, 4.85912170601302448907886571082, 5.71151921405921593743545508460, 6.81618486687405158815337366366, 7.07033947129149766441176392853, 8.098398331850279024308307632585, 9.114871554193384225253724061398, 10.24006510769264008848914452754

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