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Properties

Label 2-3225-129.110-c0-0-0
Degree $2$
Conductor $3225$
Sign $0.0805 - 0.996i$
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  + (−0.988 + 0.149i)3-s + (−0.222 − 0.974i)4-s + (−0.0747 − 0.129i)7-s + (0.955 − 0.294i)9-s + (0.365 + 0.930i)12-s + (0.142 + 1.90i)13-s + (−0.900 + 0.433i)16-s + (−1.88 − 0.582i)19-s + (0.0931 + 0.116i)21-s + (−0.900 + 0.433i)27-s + (−0.109 + 0.101i)28-s + (0.455 + 1.16i)31-s + (−0.5 − 0.866i)36-s + (−0.955 + 1.65i)37-s + (−0.425 − 1.86i)39-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)3-s + (−0.222 − 0.974i)4-s + (−0.0747 − 0.129i)7-s + (0.955 − 0.294i)9-s + (0.365 + 0.930i)12-s + (0.142 + 1.90i)13-s + (−0.900 + 0.433i)16-s + (−1.88 − 0.582i)19-s + (0.0931 + 0.116i)21-s + (−0.900 + 0.433i)27-s + (−0.109 + 0.101i)28-s + (0.455 + 1.16i)31-s + (−0.5 − 0.866i)36-s + (−0.955 + 1.65i)37-s + (−0.425 − 1.86i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0805 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0805 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3225\)    =    \(3 \cdot 5^{2} \cdot 43\)
Sign: $0.0805 - 0.996i$
Analytic conductor: \(1.60948\)
Root analytic conductor: \(1.26865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3225} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3225,\ (\ :0),\ 0.0805 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4622440560\)
\(L(\frac12)\) \(\approx\) \(0.4622440560\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.988 - 0.149i)T \)
5 \( 1 \)
43 \( 1 + (-0.365 + 0.930i)T \)
good2 \( 1 + (0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (-0.142 - 1.90i)T + (-0.988 + 0.149i)T^{2} \)
17 \( 1 + (-0.365 - 0.930i)T^{2} \)
19 \( 1 + (1.88 + 0.582i)T + (0.826 + 0.563i)T^{2} \)
23 \( 1 + (-0.0747 - 0.997i)T^{2} \)
29 \( 1 + (-0.955 - 0.294i)T^{2} \)
31 \( 1 + (-0.455 - 1.16i)T + (-0.733 + 0.680i)T^{2} \)
37 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (0.900 - 0.433i)T^{2} \)
53 \( 1 + (0.988 + 0.149i)T^{2} \)
59 \( 1 + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (0.535 - 1.36i)T + (-0.733 - 0.680i)T^{2} \)
67 \( 1 + (1.72 + 0.531i)T + (0.826 + 0.563i)T^{2} \)
71 \( 1 + (-0.0747 + 0.997i)T^{2} \)
73 \( 1 + (-0.0546 - 0.728i)T + (-0.988 + 0.149i)T^{2} \)
79 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.955 + 0.294i)T^{2} \)
89 \( 1 + (-0.955 + 0.294i)T^{2} \)
97 \( 1 + (0.367 - 1.61i)T + (-0.900 - 0.433i)T^{2} \)
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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

−9.013443527064993302483128240805, −8.608735614231784594976894398135, −7.06348340845350789686004603598, −6.63315699491147195384063790698, −6.14935164284414241699084750293, −5.10309440852591791273533360306, −4.55084173836207429540438286213, −3.91217152659000811710454587561, −2.18332580658834220416143412784, −1.29775094074102585545822435823, 0.33314794804435008458126778915, 2.04863391390898317425826984058, 3.13530700639094094688837926411, 4.07966027292631034204544669520, 4.75620346035292659630195644184, 5.82121844446223288273355210298, 6.20117100102856234149635529230, 7.32864445193065772268331026997, 7.82751927496732835726942097756, 8.490128075912121731238267594696

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