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Properties

Label 2-3675-735.47-c0-0-1
Degree $2$
Conductor $3675$
Sign $0.999 - 0.00943i$
Analytic cond. $1.83406$
Root an. cond. $1.35427$
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  + (0.916 − 0.399i)3-s + (−0.563 + 0.826i)4-s + (−0.0373 + 0.999i)7-s + (0.680 − 0.733i)9-s + (−0.185 + 0.982i)12-s + (1.05 − 1.67i)13-s + (−0.365 − 0.930i)16-s + (0.680 − 1.17i)19-s + (0.365 + 0.930i)21-s + (0.330 − 0.943i)27-s + (−0.804 − 0.593i)28-s + (−0.751 + 0.433i)31-s + (0.222 + 0.974i)36-s + (1.95 + 0.370i)37-s + (0.294 − 1.95i)39-s + ⋯
L(s)  = 1  + (0.916 − 0.399i)3-s + (−0.563 + 0.826i)4-s + (−0.0373 + 0.999i)7-s + (0.680 − 0.733i)9-s + (−0.185 + 0.982i)12-s + (1.05 − 1.67i)13-s + (−0.365 − 0.930i)16-s + (0.680 − 1.17i)19-s + (0.365 + 0.930i)21-s + (0.330 − 0.943i)27-s + (−0.804 − 0.593i)28-s + (−0.751 + 0.433i)31-s + (0.222 + 0.974i)36-s + (1.95 + 0.370i)37-s + (0.294 − 1.95i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.999 - 0.00943i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (782, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.999 - 0.00943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.674275543\)
\(L(\frac12)\) \(\approx\) \(1.674275543\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.916 + 0.399i)T \)
5 \( 1 \)
7 \( 1 + (0.0373 - 0.999i)T \)
good2 \( 1 + (0.563 - 0.826i)T^{2} \)
11 \( 1 + (-0.0747 + 0.997i)T^{2} \)
13 \( 1 + (-1.05 + 1.67i)T + (-0.433 - 0.900i)T^{2} \)
17 \( 1 + (-0.149 - 0.988i)T^{2} \)
19 \( 1 + (-0.680 + 1.17i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.149 - 0.988i)T^{2} \)
29 \( 1 + (0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.751 - 0.433i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.95 - 0.370i)T + (0.930 + 0.365i)T^{2} \)
41 \( 1 + (-0.222 + 0.974i)T^{2} \)
43 \( 1 + (-0.218 - 1.93i)T + (-0.974 + 0.222i)T^{2} \)
47 \( 1 + (-0.563 + 0.826i)T^{2} \)
53 \( 1 + (-0.930 + 0.365i)T^{2} \)
59 \( 1 + (-0.955 - 0.294i)T^{2} \)
61 \( 1 + (0.634 + 0.930i)T + (-0.365 + 0.930i)T^{2} \)
67 \( 1 + (0.481 - 1.79i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.132 - 0.0698i)T + (0.563 + 0.826i)T^{2} \)
79 \( 1 + (-1.43 - 0.826i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.433 + 0.900i)T^{2} \)
89 \( 1 + (-0.0747 - 0.997i)T^{2} \)
97 \( 1 + (1.35 - 1.35i)T - iT^{2} \)
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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

−8.522863549656046337508465277580, −8.109656917399084023957320871686, −7.56691680438396990379330761184, −6.59006662035825839396812908415, −5.72536635969291783703021881616, −4.86614178348742747736413369317, −3.86772386467138591811107399453, −2.93803415254800070828802841672, −2.73690948842777253346484621641, −1.10236126631638938951360508116, 1.26312319335992291536575007928, 2.04175039236077474813373596854, 3.58121181037508491157085912970, 4.02689976971065954944218118684, 4.64498140019931841515920006751, 5.70740478798309309243951656038, 6.48391771892330484668994056536, 7.36193946747121456899392201120, 8.041797963050173260157569751147, 8.935936617014289784861100583691

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