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Properties

Label 2-3225-129.101-c0-0-1
Degree $2$
Conductor $3225$
Sign $0.146 + 0.989i$
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  + (−0.365 + 0.930i)3-s + (−0.222 − 0.974i)4-s + (0.826 − 1.43i)7-s + (−0.733 − 0.680i)9-s + (0.988 + 0.149i)12-s + (1.21 − 0.825i)13-s + (−0.900 + 0.433i)16-s + (−0.535 + 0.496i)19-s + (1.03 + 1.29i)21-s + (0.900 − 0.433i)27-s + (−1.57 − 0.487i)28-s + (−1.23 − 0.185i)31-s + (−0.500 + 0.866i)36-s + (−0.733 − 1.26i)37-s + (0.326 + 1.42i)39-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)3-s + (−0.222 − 0.974i)4-s + (0.826 − 1.43i)7-s + (−0.733 − 0.680i)9-s + (0.988 + 0.149i)12-s + (1.21 − 0.825i)13-s + (−0.900 + 0.433i)16-s + (−0.535 + 0.496i)19-s + (1.03 + 1.29i)21-s + (0.900 − 0.433i)27-s + (−1.57 − 0.487i)28-s + (−1.23 − 0.185i)31-s + (−0.500 + 0.866i)36-s + (−0.733 − 1.26i)37-s + (0.326 + 1.42i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3225\)    =    \(3 \cdot 5^{2} \cdot 43\)
Sign: $0.146 + 0.989i$
Analytic conductor: \(1.60948\)
Root analytic conductor: \(1.26865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3225} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3225,\ (\ :0),\ 0.146 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.009988887\)
\(L(\frac12)\) \(\approx\) \(1.009988887\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.365 - 0.930i)T \)
5 \( 1 \)
43 \( 1 + (-0.988 + 0.149i)T \)
good2 \( 1 + (0.222 + 0.974i)T^{2} \)
7 \( 1 + (-0.826 + 1.43i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (-1.21 + 0.825i)T + (0.365 - 0.930i)T^{2} \)
17 \( 1 + (0.988 + 0.149i)T^{2} \)
19 \( 1 + (0.535 - 0.496i)T + (0.0747 - 0.997i)T^{2} \)
23 \( 1 + (-0.826 + 0.563i)T^{2} \)
29 \( 1 + (0.733 - 0.680i)T^{2} \)
31 \( 1 + (1.23 + 0.185i)T + (0.955 + 0.294i)T^{2} \)
37 \( 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.365 - 0.930i)T^{2} \)
59 \( 1 + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (1.88 - 0.284i)T + (0.955 - 0.294i)T^{2} \)
67 \( 1 + (1.32 - 1.22i)T + (0.0747 - 0.997i)T^{2} \)
71 \( 1 + (-0.826 - 0.563i)T^{2} \)
73 \( 1 + (-1.63 + 1.11i)T + (0.365 - 0.930i)T^{2} \)
79 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.733 + 0.680i)T^{2} \)
89 \( 1 + (0.733 + 0.680i)T^{2} \)
97 \( 1 + (-0.0332 + 0.145i)T + (-0.900 - 0.433i)T^{2} \)
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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

−8.852573314736097025404695475592, −7.998616677163578499747737430665, −7.15869564080878922227931147165, −6.09894993829768940700090313603, −5.64291695738095350929212425698, −4.75893141329372810575958181176, −4.10768579177402928812563069673, −3.46455416986465448514812851224, −1.75049090015215026943102351758, −0.66575517828098951043464987706, 1.60743429892459103801007520546, 2.36044669840462645362841521314, 3.33259690585888248047193957732, 4.48869632492552407187598412086, 5.24512173783942726093427375526, 6.11133215534779690347086891858, 6.74138451833987693493314033300, 7.64274933355345445666631171869, 8.290795822346361222398583189745, 8.817446462490753228470908305001

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