↑

Properties

Label 2-75-5.2-c2-0-5
Degree $2$
Conductor $75$
Sign $-0.973 + 0.229i$
Analytic cond. $2.04360$
Root an. cond. $1.42954$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)2-s + (−1.22 + 1.22i)3-s − 1.00i·4-s + 2.99·6-s + (−7.34 − 7.34i)7-s + (−6.12 + 6.12i)8-s − 2.99i·9-s − 18·11-s + (1.22 + 1.22i)12-s + (7.34 − 7.34i)13-s + 18i·14-s + 10.9·16-s + (4.89 + 4.89i)17-s + (−3.67 + 3.67i)18-s − 10i·19-s + ⋯
L(s)  = 1  + (−0.612 − 0.612i)2-s + (−0.408 + 0.408i)3-s − 0.250i·4-s + 0.499·6-s + (−1.04 − 1.04i)7-s + (−0.765 + 0.765i)8-s − 0.333i·9-s − 1.63·11-s + (0.102 + 0.102i)12-s + (0.565 − 0.565i)13-s + 1.28i·14-s + 0.687·16-s + (0.288 + 0.288i)17-s + (−0.204 + 0.204i)18-s − 0.526i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(2.04360\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0452253 - 0.388421i\)
\(L(\frac12)\) \(\approx\) \(0.0452253 - 0.388421i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
good2 \( 1 + (1.22 + 1.22i)T + 4iT^{2} \)
7 \( 1 + (7.34 + 7.34i)T + 49iT^{2} \)
11 \( 1 + 18T + 121T^{2} \)
13 \( 1 + (-7.34 + 7.34i)T - 169iT^{2} \)
17 \( 1 + (-4.89 - 4.89i)T + 289iT^{2} \)
19 \( 1 + 10iT - 361T^{2} \)
23 \( 1 + (-19.5 + 19.5i)T - 529iT^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 22T + 961T^{2} \)
37 \( 1 + (7.34 + 7.34i)T + 1.36e3iT^{2} \)
41 \( 1 + 18T + 1.68e3T^{2} \)
43 \( 1 + (29.3 - 29.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (44.0 + 44.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (4.89 - 4.89i)T - 2.80e3iT^{2} \)
59 \( 1 + 90iT - 3.48e3T^{2} \)
61 \( 1 - 2T + 3.72e3T^{2} \)
67 \( 1 + (44.0 + 44.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 72T + 5.04e3T^{2} \)
73 \( 1 + (-44.0 + 44.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 70iT - 6.24e3T^{2} \)
83 \( 1 + (53.8 - 53.8i)T - 6.88e3iT^{2} \)
89 \( 1 + 90iT - 7.92e3T^{2} \)
97 \( 1 + (-102. - 102. i)T + 9.40e3iT^{2} \)
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

−13.55803380555579321563664921471, −12.66974360908288497699021468758, −11.04660093779133381272355800367, −10.43272677271919347671140114634, −9.737129714682633257585266768364, −8.242542159109238893010863315701, −6.54992475316093479185306346998, −5.12847651918852031711671823561, −3.10813538946266115431392591894, −0.40958377309226192183101428220, 3.00536279826085397227320469675, 5.53765216867931320478999536059, 6.64449596392523289051848591996, 7.84828196928130944765311894468, 8.941822631590589330935127162669, 10.08845407323857217880959204922, 11.69202626635331503866525769180, 12.67557662995921968107617974357, 13.41188358960999988962852692274, 15.34217423470215891487236247701

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