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Properties

Label 2-3225-3225.866-c0-0-0
Degree $2$
Conductor $3225$
Sign $0.900 + 0.434i$
Analytic cond. $1.60948$
Root an. cond. $1.26865$
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  + (0.587 − 0.809i)2-s + (−0.978 − 0.207i)3-s + (0.743 − 0.669i)5-s + (−0.743 + 0.669i)6-s + (0.5 + 0.866i)7-s + (0.951 + 0.309i)8-s + (0.913 + 0.406i)9-s + (−0.104 − 0.994i)10-s + (−0.587 + 0.809i)11-s + (−0.564 − 0.251i)13-s + (0.994 + 0.104i)14-s + (−0.866 + 0.5i)15-s + (0.809 − 0.587i)16-s + (0.866 − 0.500i)18-s + (−0.309 − 0.951i)21-s + (0.309 + 0.951i)22-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.978 − 0.207i)3-s + (0.743 − 0.669i)5-s + (−0.743 + 0.669i)6-s + (0.5 + 0.866i)7-s + (0.951 + 0.309i)8-s + (0.913 + 0.406i)9-s + (−0.104 − 0.994i)10-s + (−0.587 + 0.809i)11-s + (−0.564 − 0.251i)13-s + (0.994 + 0.104i)14-s + (−0.866 + 0.5i)15-s + (0.809 − 0.587i)16-s + (0.866 − 0.500i)18-s + (−0.309 − 0.951i)21-s + (0.309 + 0.951i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3225\)    =    \(3 \cdot 5^{2} \cdot 43\)
Sign: $0.900 + 0.434i$
Analytic conductor: \(1.60948\)
Root analytic conductor: \(1.26865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3225} (866, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3225,\ (\ :0),\ 0.900 + 0.434i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.623238734\)
\(L(\frac12)\) \(\approx\) \(1.623238734\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (-0.743 + 0.669i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.564 + 0.251i)T + (0.669 + 0.743i)T^{2} \)
17 \( 1 + (-0.913 + 0.406i)T^{2} \)
19 \( 1 + (-0.104 + 0.994i)T^{2} \)
23 \( 1 + (-0.251 - 0.564i)T + (-0.669 + 0.743i)T^{2} \)
29 \( 1 + (-1.20 - 1.08i)T + (0.104 + 0.994i)T^{2} \)
31 \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \)
37 \( 1 + (-0.0646 - 0.614i)T + (-0.978 + 0.207i)T^{2} \)
41 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.913 - 0.406i)T^{2} \)
59 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.978 - 0.207i)T^{2} \)
67 \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \)
71 \( 1 + (1.20 + 1.08i)T + (0.104 + 0.994i)T^{2} \)
73 \( 1 + (-0.169 + 1.60i)T + (-0.978 - 0.207i)T^{2} \)
79 \( 1 + (-0.104 - 0.994i)T^{2} \)
83 \( 1 + (0.104 - 0.994i)T^{2} \)
89 \( 1 + (-0.406 - 0.913i)T + (-0.669 + 0.743i)T^{2} \)
97 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
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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

−8.873459727650339181410635492558, −7.922805274567025190091537560247, −7.33896139037422725562480768129, −6.27283695642818105650572712155, −5.44477408735796086381560338705, −4.86375077108206124696370021642, −4.51396593349020586523405641120, −2.99460782981525846048113965964, −2.11067453602662404148078974597, −1.42565699723525806912428512200, 1.03002610308982615982376612660, 2.32420871803189856847303692822, 3.73170766130896828455136317662, 4.55862430099197213890252286154, 5.26904919536397646384337793043, 5.82354647452511334112101423463, 6.55894599136215263274721361123, 7.11842742908168100337923445759, 7.69324234844648951824297034599, 8.850107765027275477363401951985

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