Properties

Label 2-75-3.2-c2-0-4
Degree $2$
Conductor $75$
Sign $0.666 - 0.745i$
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  + 2.23i·2-s + (2 − 2.23i)3-s − 1.00·4-s + (5.00 + 4.47i)6-s + 6·7-s + 6.70i·8-s + (−1.00 − 8.94i)9-s + 4.47i·11-s + (−2.00 + 2.23i)12-s − 16·13-s + 13.4i·14-s − 19·16-s − 4.47i·17-s + (20.0 − 2.23i)18-s − 2·19-s + ⋯
L(s)  = 1  + 1.11i·2-s + (0.666 − 0.745i)3-s − 0.250·4-s + (0.833 + 0.745i)6-s + 0.857·7-s + 0.838i·8-s + (−0.111 − 0.993i)9-s + 0.406i·11-s + (−0.166 + 0.186i)12-s − 1.23·13-s + 0.958i·14-s − 1.18·16-s − 0.263i·17-s + (1.11 − 0.124i)18-s − 0.105·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.666 - 0.745i$
Analytic conductor: \(2.04360\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1),\ 0.666 - 0.745i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.47931 + 0.661570i\)
\(L(\frac12)\) \(\approx\) \(1.47931 + 0.661570i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2 + 2.23i)T \)
5 \( 1 \)
good2 \( 1 - 2.23iT - 4T^{2} \)
7 \( 1 - 6T + 49T^{2} \)
11 \( 1 - 4.47iT - 121T^{2} \)
13 \( 1 + 16T + 169T^{2} \)
17 \( 1 + 4.47iT - 289T^{2} \)
19 \( 1 + 2T + 361T^{2} \)
23 \( 1 + 13.4iT - 529T^{2} \)
29 \( 1 + 31.3iT - 841T^{2} \)
31 \( 1 + 18T + 961T^{2} \)
37 \( 1 - 16T + 1.36e3T^{2} \)
41 \( 1 - 62.6iT - 1.68e3T^{2} \)
43 \( 1 + 16T + 1.84e3T^{2} \)
47 \( 1 - 49.1iT - 2.20e3T^{2} \)
53 \( 1 + 4.47iT - 2.80e3T^{2} \)
59 \( 1 - 4.47iT - 3.48e3T^{2} \)
61 \( 1 - 82T + 3.72e3T^{2} \)
67 \( 1 + 24T + 4.48e3T^{2} \)
71 \( 1 + 125. iT - 5.04e3T^{2} \)
73 \( 1 - 74T + 5.32e3T^{2} \)
79 \( 1 - 138T + 6.24e3T^{2} \)
83 \( 1 - 93.9iT - 6.88e3T^{2} \)
89 \( 1 - 107. iT - 7.92e3T^{2} \)
97 \( 1 - 166T + 9.40e3T^{2} \)
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

−14.72925744034027549497102066514, −13.75326723590149048788464616642, −12.42429911978288214491320735219, −11.35440298561855003841836180509, −9.533394723911941401829027941286, −8.167733565251724439666970532232, −7.52840572078571648059290912058, −6.41867482361789393790001923895, −4.83701565982152218221761493790, −2.29695851358570863524361320685, 2.13418391705532504473109597688, 3.62403313672418599504592721933, 5.05884099906590924491252519753, 7.37743318295281245481235974274, 8.770001270166904773054877015488, 9.907638259880481509168622235017, 10.79912101692157862322654685651, 11.71616684749326208355240428995, 12.92948695490250188622702395879, 14.17589883281362921748119557958

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