Properties

Label 2-825-11.4-c1-0-6
Degree $2$
Conductor $825$
Sign $0.440 - 0.897i$
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  + (−0.212 − 0.154i)2-s + (−0.309 + 0.951i)3-s + (−0.596 − 1.83i)4-s + (0.212 − 0.154i)6-s + (0.986 + 3.03i)7-s + (−0.318 + 0.980i)8-s + (−0.809 − 0.587i)9-s + (3.27 + 0.547i)11-s + 1.93·12-s + (−0.905 − 0.658i)13-s + (0.258 − 0.796i)14-s + (−2.90 + 2.11i)16-s + (−0.0713 + 0.0518i)17-s + (0.0810 + 0.249i)18-s + (−0.0212 + 0.0654i)19-s + ⋯
L(s)  = 1  + (−0.150 − 0.109i)2-s + (−0.178 + 0.549i)3-s + (−0.298 − 0.918i)4-s + (0.0866 − 0.0629i)6-s + (0.372 + 1.14i)7-s + (−0.112 + 0.346i)8-s + (−0.269 − 0.195i)9-s + (0.986 + 0.165i)11-s + 0.557·12-s + (−0.251 − 0.182i)13-s + (0.0691 − 0.212i)14-s + (−0.726 + 0.527i)16-s + (−0.0173 + 0.0125i)17-s + (0.0191 + 0.0588i)18-s + (−0.00487 + 0.0150i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.440 - 0.897i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.440 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.983415 + 0.612877i\)
\(L(\frac12)\) \(\approx\) \(0.983415 + 0.612877i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.27 - 0.547i)T \)
good2 \( 1 + (0.212 + 0.154i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (-0.986 - 3.03i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.905 + 0.658i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.0713 - 0.0518i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.0212 - 0.0654i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.65T + 23T^{2} \)
29 \( 1 + (-1.15 - 3.55i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.75 - 5.63i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.57 - 7.92i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.60 - 11.0i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 + (0.280 - 0.863i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.705 - 0.512i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.567 - 1.74i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.13 + 5.91i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.53T + 67T^{2} \)
71 \( 1 + (-3.77 + 2.74i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.21 - 6.80i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.640 + 0.465i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.200 + 0.145i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + (3.31 + 2.40i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.13673214216209489486550664704, −9.679949167939509811155863719592, −8.812194670731742745078164264695, −8.199302408057584499765653314272, −6.59253216451309845711903953993, −5.92898015070999894976018494230, −5.03337539545455167953999026944, −4.27455097587095417133985118543, −2.73184344621800289764851067406, −1.40342105232008346526579881594, 0.67965832646451745523761675502, 2.32546192748785917640580662200, 3.91566361282722505476714611434, 4.29555809799097869045394605660, 5.87103017637817596427562032352, 6.85813001546821051321522860402, 7.53960369553485753200555819045, 8.174958608545309597907565870532, 9.118400978981416558032964064170, 10.01346003359397021426452284479

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