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Properties

Label 2-3225-129.83-c0-0-1
Degree $2$
Conductor $3225$
Sign $0.998 - 0.0478i$
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  + (0.0747 − 0.997i)3-s + (0.623 + 0.781i)4-s + (0.733 + 1.26i)7-s + (−0.988 − 0.149i)9-s + (0.826 − 0.563i)12-s + (1.44 − 1.34i)13-s + (−0.222 + 0.974i)16-s + (−0.147 + 0.0222i)19-s + (1.32 − 0.636i)21-s + (−0.222 + 0.974i)27-s + (−0.535 + 1.36i)28-s + (−1.48 + 1.01i)31-s + (−0.499 − 0.866i)36-s + (0.988 − 1.71i)37-s + (−1.23 − 1.54i)39-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)3-s + (0.623 + 0.781i)4-s + (0.733 + 1.26i)7-s + (−0.988 − 0.149i)9-s + (0.826 − 0.563i)12-s + (1.44 − 1.34i)13-s + (−0.222 + 0.974i)16-s + (−0.147 + 0.0222i)19-s + (1.32 − 0.636i)21-s + (−0.222 + 0.974i)27-s + (−0.535 + 1.36i)28-s + (−1.48 + 1.01i)31-s + (−0.499 − 0.866i)36-s + (0.988 − 1.71i)37-s + (−1.23 − 1.54i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3225\)    =    \(3 \cdot 5^{2} \cdot 43\)
Sign: $0.998 - 0.0478i$
Analytic conductor: \(1.60948\)
Root analytic conductor: \(1.26865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3225} (2276, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3225,\ (\ :0),\ 0.998 - 0.0478i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.661323851\)
\(L(\frac12)\) \(\approx\) \(1.661323851\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0747 + 0.997i)T \)
5 \( 1 \)
43 \( 1 + (-0.826 - 0.563i)T \)
good2 \( 1 + (-0.623 - 0.781i)T^{2} \)
7 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.222 + 0.974i)T^{2} \)
13 \( 1 + (-1.44 + 1.34i)T + (0.0747 - 0.997i)T^{2} \)
17 \( 1 + (-0.826 + 0.563i)T^{2} \)
19 \( 1 + (0.147 - 0.0222i)T + (0.955 - 0.294i)T^{2} \)
23 \( 1 + (0.733 - 0.680i)T^{2} \)
29 \( 1 + (0.988 - 0.149i)T^{2} \)
31 \( 1 + (1.48 - 1.01i)T + (0.365 - 0.930i)T^{2} \)
37 \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.222 - 0.974i)T^{2} \)
53 \( 1 + (-0.0747 - 0.997i)T^{2} \)
59 \( 1 + (0.900 + 0.433i)T^{2} \)
61 \( 1 + (-0.603 - 0.411i)T + (0.365 + 0.930i)T^{2} \)
67 \( 1 + (-0.440 + 0.0663i)T + (0.955 - 0.294i)T^{2} \)
71 \( 1 + (0.733 + 0.680i)T^{2} \)
73 \( 1 + (1.21 - 1.12i)T + (0.0747 - 0.997i)T^{2} \)
79 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.988 + 0.149i)T^{2} \)
89 \( 1 + (0.988 + 0.149i)T^{2} \)
97 \( 1 + (-1.19 + 1.49i)T + (-0.222 - 0.974i)T^{2} \)
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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

−8.672712964747091573267991230023, −8.025287417040729508655708354037, −7.55255404090259140712724208972, −6.63799053534998019902881369578, −5.76852464305615200833723427245, −5.49002620491227480718280067582, −3.91925859184350243532538011408, −3.01537579521639712978676585422, −2.32198045824652574780579335581, −1.38496295985640947599073749311, 1.17260925966851518981806180694, 2.16586311880014523086946249100, 3.53469387882145236863867212973, 4.20384849476552905995295622758, 4.84775336050262329517067934534, 5.84366109849490590905612121933, 6.46371344409021151876747004058, 7.31766621453761166125253779540, 8.126736014234476317193276000973, 9.008356837724970308226594937924

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