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Properties

Label 2-3825-425.152-c0-0-1
Degree $2$
Conductor $3825$
Sign $0.790 + 0.612i$
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  + (−0.951 − 0.309i)4-s + (−0.707 + 0.707i)5-s + (−0.533 − 0.734i)11-s + (−1.39 + 0.221i)13-s + (0.809 + 0.587i)16-s + (0.891 + 0.453i)17-s + (−1.80 + 0.587i)19-s + (0.891 − 0.453i)20-s + (1.59 + 0.253i)23-s − 1.00i·25-s + (0.610 − 1.87i)29-s + (0.183 − 0.253i)41-s + (0.642 − 0.642i)43-s + (0.280 + 0.863i)44-s + i·49-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)4-s + (−0.707 + 0.707i)5-s + (−0.533 − 0.734i)11-s + (−1.39 + 0.221i)13-s + (0.809 + 0.587i)16-s + (0.891 + 0.453i)17-s + (−1.80 + 0.587i)19-s + (0.891 − 0.453i)20-s + (1.59 + 0.253i)23-s − 1.00i·25-s + (0.610 − 1.87i)29-s + (0.183 − 0.253i)41-s + (0.642 − 0.642i)43-s + (0.280 + 0.863i)44-s + i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3825\)    =    \(3^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(1.90892\)
Root analytic conductor: \(1.38163\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3825} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3825,\ (\ :0),\ 0.790 + 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6341615867\)
\(L(\frac12)\) \(\approx\) \(0.6341615867\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-0.891 - 0.453i)T \)
good2 \( 1 + (0.951 + 0.309i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (0.533 + 0.734i)T + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.39 - 0.221i)T + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-1.59 - 0.253i)T + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (-0.610 + 1.87i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (-0.183 + 0.253i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
47 \( 1 + (0.587 - 0.809i)T^{2} \)
53 \( 1 + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (-1.69 - 0.550i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.951 + 0.309i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.587 + 0.809i)T^{2} \)
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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

−8.349341014674585280496929916622, −8.066984763642318044443020246866, −7.22262060150610012287790500054, −6.31401330615974811029796830448, −5.60763799089179482614184574810, −4.68589755877005296526400177836, −4.07715210377460087189566704009, −3.18706261321946963149943560944, −2.24728301910177354399500027723, −0.52226518193599434735136459803, 0.859288152410451607436130641431, 2.47882538107399024248976653354, 3.36967702317251878144830711435, 4.40985641783056078422859582471, 4.95335633736896599337116870168, 5.26229865758401181309241753957, 6.82592967441177549415847003178, 7.37844354469440849628903005475, 8.088132692002467425587262055021, 8.750668122529500689204847718325

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