↑

Properties

Label 2-375-25.9-c1-0-8
Degree $2$
Conductor $375$
Sign $0.894 + 0.447i$
Analytic cond. $2.99439$
Root an. cond. $1.73043$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.640 − 0.881i)2-s + (−0.951 + 0.309i)3-s + (0.251 + 0.772i)4-s + (−0.336 + 1.03i)6-s − 3.08i·7-s + (2.91 + 0.947i)8-s + (0.809 − 0.587i)9-s + (0.929 + 0.674i)11-s + (−0.477 − 0.657i)12-s + (2.39 + 3.30i)13-s + (−2.72 − 1.97i)14-s + (1.38 − 1.00i)16-s + (4.40 + 1.42i)17-s − 1.08i·18-s + (1.84 − 5.67i)19-s + ⋯
L(s)  = 1  + (0.452 − 0.623i)2-s + (−0.549 + 0.178i)3-s + (0.125 + 0.386i)4-s + (−0.137 + 0.423i)6-s − 1.16i·7-s + (1.03 + 0.334i)8-s + (0.269 − 0.195i)9-s + (0.280 + 0.203i)11-s + (−0.137 − 0.189i)12-s + (0.665 + 0.915i)13-s + (−0.727 − 0.528i)14-s + (0.346 − 0.252i)16-s + (1.06 + 0.346i)17-s − 0.256i·18-s + (0.423 − 1.30i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61346 - 0.380705i\)
\(L(\frac12)\) \(\approx\) \(1.61346 - 0.380705i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 \)
good2 \( 1 + (-0.640 + 0.881i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + 3.08iT - 7T^{2} \)
11 \( 1 + (-0.929 - 0.674i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.39 - 3.30i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.40 - 1.42i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.84 + 5.67i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.36 + 1.88i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.63 - 5.02i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.182 - 0.560i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (6.70 + 9.22i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.67 - 5.57i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.42iT - 43T^{2} \)
47 \( 1 + (5.75 - 1.86i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.08 - 1.00i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.57 + 1.87i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (11.1 + 8.07i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-3.00 - 0.976i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.99 - 6.14i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.23 - 5.83i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.81 - 11.7i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (11.7 + 3.82i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-0.877 - 0.637i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (4.30 - 1.39i)T + (78.4 - 57.0i)T^{2} \)
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

−11.22236719564872731924641655735, −10.78089219681687160678943250613, −9.786754197917481038969197590807, −8.557755045073342377898167415569, −7.29408493383762776232845371798, −6.70977104171002808298939207233, −5.09629082779778962500317757487, −4.16772216517020672434491177052, −3.29537418680095142477141027200, −1.42684702139889946698274088726, 1.46733753727911589143139380231, 3.36464449007159669428320124027, 5.02849631624172131664204867234, 5.72248263295790591972342055858, 6.29436403225727588721995997770, 7.54893708238301029053357610154, 8.447616559738121867828173948703, 9.806525995212239532396019160050, 10.47444109652003779018802302961, 11.68046052069447325072999026694

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