↑

Properties

Label 2-3675-735.248-c0-0-1
Degree $2$
Conductor $3675$
Sign $-0.256 - 0.966i$
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  + (0.652 + 0.757i)3-s + (−0.680 + 0.733i)4-s + (0.804 + 0.593i)7-s + (−0.149 + 0.988i)9-s + (−0.999 − 0.0373i)12-s + (1.64 + 0.185i)13-s + (−0.0747 − 0.997i)16-s + (−0.149 − 0.258i)19-s + (0.0747 + 0.997i)21-s + (−0.846 + 0.532i)27-s + (−0.982 + 0.185i)28-s + (1.68 + 0.974i)31-s + (−0.623 − 0.781i)36-s + (−0.0220 + 0.589i)37-s + (0.930 + 1.36i)39-s + ⋯
L(s)  = 1  + (0.652 + 0.757i)3-s + (−0.680 + 0.733i)4-s + (0.804 + 0.593i)7-s + (−0.149 + 0.988i)9-s + (−0.999 − 0.0373i)12-s + (1.64 + 0.185i)13-s + (−0.0747 − 0.997i)16-s + (−0.149 − 0.258i)19-s + (0.0747 + 0.997i)21-s + (−0.846 + 0.532i)27-s + (−0.982 + 0.185i)28-s + (1.68 + 0.974i)31-s + (−0.623 − 0.781i)36-s + (−0.0220 + 0.589i)37-s + (0.930 + 1.36i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.256 - 0.966i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1718, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.256 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.597179625\)
\(L(\frac12)\) \(\approx\) \(1.597179625\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.652 - 0.757i)T \)
5 \( 1 \)
7 \( 1 + (-0.804 - 0.593i)T \)
good2 \( 1 + (0.680 - 0.733i)T^{2} \)
11 \( 1 + (-0.955 + 0.294i)T^{2} \)
13 \( 1 + (-1.64 - 0.185i)T + (0.974 + 0.222i)T^{2} \)
17 \( 1 + (0.563 - 0.826i)T^{2} \)
19 \( 1 + (0.149 + 0.258i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.563 - 0.826i)T^{2} \)
29 \( 1 + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (-1.68 - 0.974i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.0220 - 0.589i)T + (-0.997 - 0.0747i)T^{2} \)
41 \( 1 + (0.623 - 0.781i)T^{2} \)
43 \( 1 + (0.516 + 1.47i)T + (-0.781 + 0.623i)T^{2} \)
47 \( 1 + (-0.680 + 0.733i)T^{2} \)
53 \( 1 + (0.997 - 0.0747i)T^{2} \)
59 \( 1 + (-0.365 + 0.930i)T^{2} \)
61 \( 1 + (0.925 + 0.997i)T + (-0.0747 + 0.997i)T^{2} \)
67 \( 1 + (0.516 + 1.92i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (1.75 + 0.764i)T + (0.680 + 0.733i)T^{2} \)
79 \( 1 + (1.26 - 0.733i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.974 - 0.222i)T^{2} \)
89 \( 1 + (-0.955 - 0.294i)T^{2} \)
97 \( 1 + (-0.516 - 0.516i)T + iT^{2} \)
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

−8.837573941855350401834262398774, −8.305132295895854993271481589722, −7.917831969868768028173472890802, −6.78411076117126227260982582057, −5.74996678227219330985373623279, −4.87073745290995871512697459938, −4.36362378867545733456804756139, −3.48464880946752467266155533057, −2.82660939931370062027779675631, −1.59949529600465407817098029504, 1.01743875921605557470294289125, 1.58606958889050531313056609358, 2.89135737788635332826758254092, 4.01064744354698518160516960983, 4.46093512785393593434076679037, 5.74135288452990941183445817617, 6.15801334911581279107363973039, 7.07932219980072563504678322907, 7.991489900187035351804765612276, 8.425570832301907537875164105015

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