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Properties

Label 2-825-11.9-c1-0-0
Degree $2$
Conductor $825$
Sign $-0.778 + 0.627i$
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  + (−0.359 + 1.10i)2-s + (0.809 − 0.587i)3-s + (0.525 + 0.381i)4-s + (0.359 + 1.10i)6-s + (−3.46 − 2.51i)7-s + (−2.49 + 1.80i)8-s + (0.309 − 0.951i)9-s + (−3.15 − 1.00i)11-s + 0.649·12-s + (−1.59 + 4.91i)13-s + (4.03 − 2.92i)14-s + (−0.704 − 2.16i)16-s + (−1.54 − 4.75i)17-s + (0.940 + 0.683i)18-s + (−4.53 + 3.29i)19-s + ⋯
L(s)  = 1  + (−0.253 + 0.781i)2-s + (0.467 − 0.339i)3-s + (0.262 + 0.190i)4-s + (0.146 + 0.451i)6-s + (−1.31 − 0.952i)7-s + (−0.880 + 0.639i)8-s + (0.103 − 0.317i)9-s + (−0.952 − 0.304i)11-s + 0.187·12-s + (−0.442 + 1.36i)13-s + (1.07 − 0.782i)14-s + (−0.176 − 0.541i)16-s + (−0.374 − 1.15i)17-s + (0.221 + 0.161i)18-s + (−1.03 + 0.755i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.778 + 0.627i$
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: \((2,\ 825,\ (\ :1/2),\ -0.778 + 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0207455 - 0.0587581i\)
\(L(\frac12)\) \(\approx\) \(0.0207455 - 0.0587581i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.15 + 1.00i)T \)
good2 \( 1 + (0.359 - 1.10i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (3.46 + 2.51i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.59 - 4.91i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.54 + 4.75i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.53 - 3.29i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 0.219T + 23T^{2} \)
29 \( 1 + (5.19 + 3.77i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.874 - 2.69i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.17 - 2.30i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.74 - 3.44i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 8.90T + 43T^{2} \)
47 \( 1 + (0.192 - 0.139i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.783 - 2.41i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.36 - 4.62i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.50 + 4.62i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + (3.08 + 9.49i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-11.7 - 8.55i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.47 + 7.61i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.87 + 11.9i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 2.56T + 89T^{2} \)
97 \( 1 + (0.621 - 1.91i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.60695600904367603140622602223, −9.619112528344002577908572501676, −9.000407314536086889837892905602, −7.945245557128002767097222436056, −7.24974552789230020164022092684, −6.69089939801187540202316824865, −5.91008050419141749786892124521, −4.39735155938394862145143882000, −3.25141994012966842411687633599, −2.28502476274412261980154198528, 0.02719591954086148918470482113, 2.25584792631942894207701098680, 2.78868754991763172346862368031, 3.79301327269138391813260019062, 5.37271348034500299477137093223, 6.08827241099219364065705516013, 7.13529612847013755785388005117, 8.309926985757658064109615284327, 9.133966615553492761463294011158, 9.805008782992232846361206361657

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