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Properties

Label 2-3825-153.101-c0-0-1
Degree $2$
Conductor $3825$
Sign $0.342 + 0.939i$
Analytic cond. $1.90892$
Root an. cond. $1.38163$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (1.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s − 0.999·12-s + (−0.499 + 0.866i)16-s + 17-s + 19-s + (1.5 − 0.866i)21-s + (0.5 + 0.866i)23-s − 0.999·27-s − 1.73i·28-s + (−0.499 + 0.866i)36-s − 1.73i·37-s + (0.499 + 0.866i)48-s + (1 + 1.73i)49-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (1.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s − 0.999·12-s + (−0.499 + 0.866i)16-s + 17-s + 19-s + (1.5 − 0.866i)21-s + (0.5 + 0.866i)23-s − 0.999·27-s − 1.73i·28-s + (−0.499 + 0.866i)36-s − 1.73i·37-s + (0.499 + 0.866i)48-s + (1 + 1.73i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3825\)    =    \(3^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(1.90892\)
Root analytic conductor: \(1.38163\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3825} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3825,\ (\ :0),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.678890712\)
\(L(\frac12)\) \(\approx\) \(1.678890712\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
17 \( 1 - T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - 1.73iT - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−8.522974979976101771396119245979, −7.78166950221820073782574933140, −7.32375744820855210738854006924, −6.12792237955589470019045381166, −5.49890644192248517237878613523, −5.05389502217405582218877224345, −3.90671635786625681010683058773, −2.79060490272791069552515271194, −1.76320796766063644168408178155, −1.16536305297404719656082042812, 1.32321784022189186835471082234, 2.73660952925442040489742480872, 3.47181905940358245923432407272, 4.28942495713241278839922304222, 4.83411339232792973825310485920, 5.41016698172165325190631812495, 6.88473829866581444686135386446, 7.75810522965743948890678049139, 8.029061443487869447429976514249, 8.665804601129290885433619126862

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