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Properties

Label 2-825-11.9-c1-0-36
Degree $2$
Conductor $825$
Sign $0.605 - 0.795i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 2.48i)2-s + (−0.809 + 0.587i)3-s + (−3.92 − 2.85i)4-s + (0.809 + 2.48i)6-s + (−0.809 − 0.587i)7-s + (−6.04 + 4.39i)8-s + (0.309 − 0.951i)9-s + (−3.30 − 0.224i)11-s + 4.85·12-s + (−0.0729 + 0.224i)13-s + (−2.11 + 1.53i)14-s + (3.04 + 9.37i)16-s + (0.354 + 1.08i)17-s + (−2.11 − 1.53i)18-s + (−4.73 + 3.44i)19-s + ⋯
L(s)  = 1  + (0.572 − 1.76i)2-s + (−0.467 + 0.339i)3-s + (−1.96 − 1.42i)4-s + (0.330 + 1.01i)6-s + (−0.305 − 0.222i)7-s + (−2.13 + 1.55i)8-s + (0.103 − 0.317i)9-s + (−0.997 − 0.0676i)11-s + 1.40·12-s + (−0.0202 + 0.0622i)13-s + (−0.566 + 0.411i)14-s + (0.761 + 2.34i)16-s + (0.0858 + 0.264i)17-s + (−0.499 − 0.362i)18-s + (−1.08 + 0.789i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.605 - 0.795i$
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: \(1\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.30 + 0.224i)T \)
good2 \( 1 + (-0.809 + 2.48i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.0729 - 0.224i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.354 - 1.08i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.236T + 23T^{2} \)
29 \( 1 + (-4.85 - 3.52i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.88 - 5.79i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.04 + 3.66i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.190 - 0.138i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 6.70T + 43T^{2} \)
47 \( 1 + (8.16 - 5.93i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.118 + 0.363i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.97 + 4.33i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.57 + 10.9i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 1.85T + 67T^{2} \)
71 \( 1 + (-3.19 - 9.82i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.61 + 3.35i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.39 + 10.4i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.454 + 1.40i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 8.23T + 89T^{2} \)
97 \( 1 + (2.42 - 7.46i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.982718852717383621102902107588, −9.087314171876098426105435916388, −8.113152125615874613717535416838, −6.56315553472296464045941603007, −5.49474566344415066909173765969, −4.74770029685664654720755678031, −3.80306157794372654404007056072, −2.92257934806023685130691371979, −1.64985787875352739664082176635, 0, 2.73130761677589020497015689306, 4.23170214436569561634896961683, 5.02362418426370079843457011831, 5.85987782458731005950713473872, 6.53529626151337167468515020117, 7.34387920484458403009441156146, 8.083127605216193115645183043771, 8.845663864004264616295803561125, 9.887529383112175425688023013492

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