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Properties

Label 2-3225-129.53-c0-0-0
Degree $2$
Conductor $3225$
Sign $-0.588 - 0.808i$
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  + (0.733 + 0.680i)3-s + (−0.900 + 0.433i)4-s + (0.365 + 0.632i)7-s + (0.0747 + 0.997i)9-s + (−0.955 − 0.294i)12-s + (−0.0546 + 0.139i)13-s + (0.623 − 0.781i)16-s + (−0.109 + 1.46i)19-s + (−0.162 + 0.712i)21-s + (−0.623 + 0.781i)27-s + (−0.603 − 0.411i)28-s + (−0.425 − 0.131i)31-s + (−0.500 − 0.866i)36-s + (0.0747 − 0.129i)37-s + (−0.134 + 0.0648i)39-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)3-s + (−0.900 + 0.433i)4-s + (0.365 + 0.632i)7-s + (0.0747 + 0.997i)9-s + (−0.955 − 0.294i)12-s + (−0.0546 + 0.139i)13-s + (0.623 − 0.781i)16-s + (−0.109 + 1.46i)19-s + (−0.162 + 0.712i)21-s + (−0.623 + 0.781i)27-s + (−0.603 − 0.411i)28-s + (−0.425 − 0.131i)31-s + (−0.500 − 0.866i)36-s + (0.0747 − 0.129i)37-s + (−0.134 + 0.0648i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3225\)    =    \(3 \cdot 5^{2} \cdot 43\)
Sign: $-0.588 - 0.808i$
Analytic conductor: \(1.60948\)
Root analytic conductor: \(1.26865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3225} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3225,\ (\ :0),\ -0.588 - 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.214608804\)
\(L(\frac12)\) \(\approx\) \(1.214608804\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.733 - 0.680i)T \)
5 \( 1 \)
43 \( 1 + (0.955 - 0.294i)T \)
good2 \( 1 + (0.900 - 0.433i)T^{2} \)
7 \( 1 + (-0.365 - 0.632i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.623 - 0.781i)T^{2} \)
13 \( 1 + (0.0546 - 0.139i)T + (-0.733 - 0.680i)T^{2} \)
17 \( 1 + (-0.955 - 0.294i)T^{2} \)
19 \( 1 + (0.109 - 1.46i)T + (-0.988 - 0.149i)T^{2} \)
23 \( 1 + (-0.365 + 0.930i)T^{2} \)
29 \( 1 + (-0.0747 + 0.997i)T^{2} \)
31 \( 1 + (0.425 + 0.131i)T + (0.826 + 0.563i)T^{2} \)
37 \( 1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.733 - 0.680i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (-1.57 + 0.487i)T + (0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.0931 - 1.24i)T + (-0.988 - 0.149i)T^{2} \)
71 \( 1 + (-0.365 - 0.930i)T^{2} \)
73 \( 1 + (0.698 - 1.77i)T + (-0.733 - 0.680i)T^{2} \)
79 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.0747 - 0.997i)T^{2} \)
89 \( 1 + (-0.0747 - 0.997i)T^{2} \)
97 \( 1 + (1.78 + 0.858i)T + (0.623 + 0.781i)T^{2} \)
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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

−8.900137385874153777341666055841, −8.485259926958563309658623412990, −7.950013014070636560333702831071, −7.11122421905790845632293533721, −5.75665210576986739201011746812, −5.20740059586525365190289684488, −4.29884762053546129305052230127, −3.72206964082312607990596782224, −2.83307345205857971148302072094, −1.75486181788475837573644400391, 0.70519805839037107644447726491, 1.77469592019896743947261458544, 2.94449698071429987735574353065, 3.89592769400449044841009865719, 4.64208456284898502059995022261, 5.47001170889089381833715262054, 6.49183485613103063268842925507, 7.15861350900377535998196481113, 7.925252434208261826513241440044, 8.599597733678769634669058664719

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