↑

Properties

Label 2-3675-735.563-c0-0-0
Degree $2$
Conductor $3675$
Sign $0.209 - 0.977i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.209 - 0.977i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (2768, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.209 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3805817492\)
\(L(\frac12)\) \(\approx\) \(0.3805817492\)
\(L(1)\) 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 \( 1 + (-1.35 - 1.35i)T + iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.891484145942701550001869256834, −8.032299996859403119843940263090, −7.45545399302922426960006876167, −6.37243788319000850568343244739, −5.70737482042355045932462463254, −5.30461820303423186280525428192, −4.72440287480403069637853527934, −3.39893950911156804330920174261, −2.22254091787533766586539736997, −1.18904750113611290374054897516, 0.27565822470788486415140637420, 1.91286413227104100775233703651, 3.40490662549113478926522604960, 3.98017318891792532093163060582, 4.88061729195474066768387671150, 5.09349230207766783834354819361, 6.59415978879834402315142671709, 7.09750899166324134529454810847, 7.49835915616156996803917142550, 8.780107202400301091856885869881

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