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Properties

Label 2-825-275.86-c1-0-40
Degree $2$
Conductor $825$
Sign $0.760 - 0.649i$
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  + (0.309 + 0.951i)3-s + (1.61 − 1.17i)4-s + (0.690 + 2.12i)5-s + (3.73 + 2.71i)7-s + (−0.809 + 0.587i)9-s + (2.19 − 2.48i)11-s + (1.61 + 1.17i)12-s + (3.73 − 2.71i)13-s + (−1.80 + 1.31i)15-s + (1.23 − 3.80i)16-s + (−1.42 + 1.03i)17-s + (−6.04 − 4.39i)19-s + (3.61 + 2.62i)20-s + (−1.42 + 4.39i)21-s + (−1.92 + 1.40i)23-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s + (0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (1.41 + 1.02i)7-s + (−0.269 + 0.195i)9-s + (0.660 − 0.750i)11-s + (0.467 + 0.339i)12-s + (1.03 − 0.752i)13-s + (−0.467 + 0.339i)15-s + (0.309 − 0.951i)16-s + (−0.346 + 0.251i)17-s + (−1.38 − 1.00i)19-s + (0.809 + 0.587i)20-s + (−0.311 + 0.958i)21-s + (−0.401 + 0.291i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.760 - 0.649i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.760 - 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29441 + 0.845903i\)
\(L(\frac12)\) \(\approx\) \(2.29441 + 0.845903i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.690 - 2.12i)T \)
11 \( 1 + (-2.19 + 2.48i)T \)
good2 \( 1 + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (-3.73 - 2.71i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-3.73 + 2.71i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.42 - 1.03i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (6.04 + 4.39i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.92 - 1.40i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (4.61 - 3.35i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (1.78 + 5.48i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 - 1.76T + 41T^{2} \)
43 \( 1 - 1.76T + 43T^{2} \)
47 \( 1 + (-1.11 - 3.44i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.28 + 5.29i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.39 + 7.38i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.19 - 2.31i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.690 + 2.12i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + 2.52T + 71T^{2} \)
73 \( 1 - 5.70T + 73T^{2} \)
79 \( 1 + (-9.78 - 7.10i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (9.78 - 7.10i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (10.2 - 7.46i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-5.35 - 3.88i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.78280234815469004473962074659, −9.451651582436327710684552727113, −8.684813800996140561772953339811, −7.87974763438558497304675317501, −6.64277381519481764208462653994, −5.90594442629605209396769151066, −5.24702642586929971226439688045, −3.80284173617101765334024278814, −2.60902084660006798921724711411, −1.72892718357497994306824864663, 1.52457450253782426516071951460, 1.92842095892573699555912674571, 3.97495364204366857557835664364, 4.41835457874730489236940159864, 5.95259097990392430900725268471, 6.77368049153225178107741899994, 7.67886384066213025642617475039, 8.291688602062795139077523309332, 8.984016384470568630002457184552, 10.27534496640855353450427959913

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