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Properties

Label 2-825-11.3-c1-0-34
Degree $2$
Conductor $825$
Sign $-0.100 + 0.994i$
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  + (2.20 − 1.60i)2-s + (0.309 + 0.951i)3-s + (1.67 − 5.16i)4-s + (2.20 + 1.60i)6-s + (−0.150 + 0.462i)7-s + (−2.88 − 8.89i)8-s + (−0.809 + 0.587i)9-s + (3.06 + 1.26i)11-s + 5.43·12-s + (2.92 − 2.12i)13-s + (0.409 + 1.26i)14-s + (−11.8 − 8.59i)16-s + (−2.94 − 2.14i)17-s + (−0.842 + 2.59i)18-s + (−0.504 − 1.55i)19-s + ⋯
L(s)  = 1  + (1.55 − 1.13i)2-s + (0.178 + 0.549i)3-s + (0.839 − 2.58i)4-s + (0.900 + 0.654i)6-s + (−0.0568 + 0.174i)7-s + (−1.02 − 3.14i)8-s + (−0.269 + 0.195i)9-s + (0.924 + 0.381i)11-s + 1.56·12-s + (0.810 − 0.588i)13-s + (0.109 + 0.336i)14-s + (−2.95 − 2.14i)16-s + (−0.714 − 0.519i)17-s + (−0.198 + 0.611i)18-s + (−0.115 − 0.355i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.100 + 0.994i$
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: \((2,\ 825,\ (\ :1/2),\ -0.100 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.62885 - 2.90906i\)
\(L(\frac12)\) \(\approx\) \(2.62885 - 2.90906i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.06 - 1.26i)T \)
good2 \( 1 + (-2.20 + 1.60i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.150 - 0.462i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.92 + 2.12i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.94 + 2.14i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.504 + 1.55i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.54T + 23T^{2} \)
29 \( 1 + (-0.326 + 1.00i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.93 - 3.58i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.65 - 8.17i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.39 - 10.4i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.41T + 43T^{2} \)
47 \( 1 + (1.61 + 4.95i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.20 - 0.875i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.46 - 7.58i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-8.78 - 6.38i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 0.432T + 67T^{2} \)
71 \( 1 + (4.86 + 3.53i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.359 + 1.10i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.57 + 2.60i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.57 - 1.86i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 1.38T + 89T^{2} \)
97 \( 1 + (1.13 - 0.822i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.30066172168846589229082992854, −9.490627864998833629862411417318, −8.689736847133377902404262204743, −6.95443826901507375215120931241, −6.17105838881627345728777454064, −5.15178536324223104316330929477, −4.45096256549731660216401028039, −3.52587235889781025805282356891, −2.72436009367773811788371386498, −1.37098665775371471921761306976, 2.05710247869621753091971931670, 3.57046656757267854981545944903, 4.02258712109308850938013917379, 5.30456819679920883084413702343, 6.19507552133500140926860685554, 6.74745170596026342260478291099, 7.48871528209337333996828013425, 8.525598144341531885677626004844, 9.059181842571559348521869068435, 10.95360364675247013597436403690

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