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Properties

Label 2-825-55.9-c1-0-25
Degree $2$
Conductor $825$
Sign $-0.415 + 0.909i$
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  + (−1.10 − 0.359i)2-s + (−0.587 − 0.809i)3-s + (−0.525 − 0.381i)4-s + (0.359 + 1.10i)6-s + (2.51 − 3.46i)7-s + (1.80 + 2.49i)8-s + (−0.309 + 0.951i)9-s + (−3.15 − 1.00i)11-s + 0.649i·12-s + (4.91 + 1.59i)13-s + (−4.03 + 2.92i)14-s + (−0.704 − 2.16i)16-s + (4.75 − 1.54i)17-s + (0.683 − 0.940i)18-s + (4.53 − 3.29i)19-s + ⋯
L(s)  = 1  + (−0.781 − 0.253i)2-s + (−0.339 − 0.467i)3-s + (−0.262 − 0.190i)4-s + (0.146 + 0.451i)6-s + (0.952 − 1.31i)7-s + (0.639 + 0.880i)8-s + (−0.103 + 0.317i)9-s + (−0.952 − 0.304i)11-s + 0.187i·12-s + (1.36 + 0.442i)13-s + (−1.07 + 0.782i)14-s + (−0.176 − 0.541i)16-s + (1.15 − 0.374i)17-s + (0.161 − 0.221i)18-s + (1.03 − 0.755i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.415 + 0.909i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.415 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.496052 - 0.771822i\)
\(L(\frac12)\) \(\approx\) \(0.496052 - 0.771822i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 \)
11 \( 1 + (3.15 + 1.00i)T \)
good2 \( 1 + (1.10 + 0.359i)T + (1.61 + 1.17i)T^{2} \)
7 \( 1 + (-2.51 + 3.46i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-4.91 - 1.59i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.75 + 1.54i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.53 + 3.29i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.219iT - 23T^{2} \)
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 \( 1 + (2.30 - 3.17i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.74 - 3.44i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.90iT - 43T^{2} \)
47 \( 1 + (0.139 + 0.192i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.41 - 0.783i)T + (42.8 + 31.1i)T^{2} \)
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 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 + (3.08 + 9.49i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.55 + 11.7i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.47 - 7.61i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (11.9 - 3.87i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 2.56T + 89T^{2} \)
97 \( 1 + (1.91 + 0.621i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.19357791307167745147224696493, −9.068141216657320252940887094950, −8.151156155606258050247114088942, −7.68736602991196236500536651905, −6.71484898341740096150520149514, −5.33756416974834285345756284899, −4.79194448732210058163634780209, −3.35345276770681658968475696374, −1.55739167475236386281545529736, −0.77607161498215799968805377563, 1.32739159184348250666765241466, 3.04884892329357990979297642765, 4.24755619754969152255858346517, 5.41146597385423407323658575553, 5.86193040062459590894940879181, 7.42640183616929493910880585960, 8.274014195209728505393395249837, 8.538751878827395978091607799878, 9.695918811683006412889637014523, 10.23533254738376587026703479058

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