Properties

Label 2-2925-117.38-c0-0-1
Degree $2$
Conductor $2925$
Sign $0.642 - 0.766i$
Analytic cond. $1.45976$
Root an. cond. $1.20820$
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  + 3-s + (0.5 + 0.866i)4-s + 9-s + (0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + 1.73i·17-s + (−1.5 + 0.866i)23-s + 27-s + (1.5 + 0.866i)29-s + (0.5 + 0.866i)36-s + (−0.5 − 0.866i)39-s + (1 − 1.73i)43-s + (−0.499 + 0.866i)48-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + 3-s + (0.5 + 0.866i)4-s + 9-s + (0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.499 + 0.866i)16-s + 1.73i·17-s + (−1.5 + 0.866i)23-s + 27-s + (1.5 + 0.866i)29-s + (0.5 + 0.866i)36-s + (−0.5 − 0.866i)39-s + (1 − 1.73i)43-s + (−0.499 + 0.866i)48-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (1676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.951858078\)
\(L(\frac12)\) \(\approx\) \(1.951858078\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.652413244334396453544107483757, −8.312374516755770733425426926118, −7.73503466082006036443695530623, −6.96838665869194703548953374107, −6.20777470872312083621191869065, −5.10014227738254434226044130620, −3.87156994043140486267481632491, −3.56335907794005351837963543553, −2.49831245392009793428463772958, −1.74083266992747888707149059598, 1.16407494743949634103978667390, 2.43476575950368154038989713057, 2.74032081412975021635178061459, 4.32192066677180653193422690428, 4.68804668415531114189549644890, 5.94119926150063309653908041131, 6.64540413559197328850293174871, 7.35271946575011852512331160784, 8.012099382562616019296108178265, 9.011803129509337660087032524685

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