↑

Properties

Label 2-3150-15.14-c2-0-23
Degree $2$
Conductor $3150$
Sign $-0.151 - 0.988i$
Analytic cond. $85.8312$
Root an. cond. $9.26451$
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  + 1.41·2-s + 2.00·4-s + 2.64i·7-s + 2.82·8-s + 17.7i·11-s − 2.58i·13-s + 3.74i·14-s + 4.00·16-s + 25.8·17-s − 20·19-s + 25.1i·22-s + 17.7·23-s − 3.65i·26-s + 5.29i·28-s − 11.9i·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.500·4-s + 0.377i·7-s + 0.353·8-s + 1.61i·11-s − 0.198i·13-s + 0.267i·14-s + 0.250·16-s + 1.52·17-s − 1.05·19-s + 1.14i·22-s + 0.773·23-s − 0.140i·26-s + 0.188i·28-s − 0.410i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.151 - 0.988i$
Analytic conductor: \(85.8312\)
Root analytic conductor: \(9.26451\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{3150} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3150,\ (\ :1),\ -0.151 - 0.988i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.946790257\)
\(L(\frac12)\) \(\approx\) \(2.946790257\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - 2.64iT \)
good11 \( 1 - 17.7iT - 121T^{2} \)
13 \( 1 + 2.58iT - 169T^{2} \)
17 \( 1 - 25.8T + 289T^{2} \)
19 \( 1 + 20T + 361T^{2} \)
23 \( 1 - 17.7T + 529T^{2} \)
29 \( 1 + 11.9iT - 841T^{2} \)
31 \( 1 + 17.1T + 961T^{2} \)
37 \( 1 - 38iT - 1.36e3T^{2} \)
41 \( 1 + 15.7iT - 1.68e3T^{2} \)
43 \( 1 - 43.4iT - 1.84e3T^{2} \)
47 \( 1 - 16.9T + 2.20e3T^{2} \)
53 \( 1 + 85.5T + 2.80e3T^{2} \)
59 \( 1 + 1.64iT - 3.48e3T^{2} \)
61 \( 1 - 100.T + 3.72e3T^{2} \)
67 \( 1 - 36.6iT - 4.48e3T^{2} \)
71 \( 1 - 17.7iT - 5.04e3T^{2} \)
73 \( 1 + 28.9iT - 5.32e3T^{2} \)
79 \( 1 + 118.T + 6.24e3T^{2} \)
83 \( 1 + 120.T + 6.88e3T^{2} \)
89 \( 1 - 139. iT - 7.92e3T^{2} \)
97 \( 1 - 44.4iT - 9.40e3T^{2} \)
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

−8.607340842067623115161535245181, −7.81240655381805374445371314867, −7.14798639451125331165585531549, −6.41040904576472821792124391725, −5.54734691960228224194743579337, −4.86341996211258518774538664650, −4.14102771140271101441567629541, −3.14581988096862544055459023119, −2.27464183459988646904685657743, −1.31208540476813194538984957074, 0.49621489085441696905330482240, 1.60372400648997039975453940284, 2.91805918035537025698396218436, 3.52011903831808764116585538220, 4.31674571756032453299901920548, 5.40529256210918294754442513327, 5.82762705588677279554390256315, 6.71902326722015688552977778450, 7.44622162422617084568059574263, 8.284307023900877308570056261862

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