Properties

Label 2-1500-25.9-c1-0-12
Degree $2$
Conductor $1500$
Sign $-0.776 + 0.630i$
Analytic cond. $11.9775$
Root an. cond. $3.46086$
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  + (0.951 − 0.309i)3-s + 1.04i·7-s + (0.809 − 0.587i)9-s + (−5.08 − 3.69i)11-s + (−0.591 − 0.814i)13-s + (−4.46 − 1.44i)17-s + (1.84 − 5.68i)19-s + (0.323 + 0.995i)21-s + (−4.73 + 6.51i)23-s + (0.587 − 0.809i)27-s + (−2.13 − 6.57i)29-s + (−2.94 + 9.05i)31-s + (−5.98 − 1.94i)33-s + (−4.52 − 6.22i)37-s + (−0.814 − 0.591i)39-s + ⋯
L(s)  = 1  + (0.549 − 0.178i)3-s + 0.395i·7-s + (0.269 − 0.195i)9-s + (−1.53 − 1.11i)11-s + (−0.164 − 0.225i)13-s + (−1.08 − 0.351i)17-s + (0.423 − 1.30i)19-s + (0.0705 + 0.217i)21-s + (−0.986 + 1.35i)23-s + (0.113 − 0.155i)27-s + (−0.396 − 1.22i)29-s + (−0.528 + 1.62i)31-s + (−1.04 − 0.338i)33-s + (−0.743 − 1.02i)37-s + (−0.130 − 0.0947i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.776 + 0.630i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.776 + 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8454732665\)
\(L(\frac12)\) \(\approx\) \(0.8454732665\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 \)
good7 \( 1 - 1.04iT - 7T^{2} \)
11 \( 1 + (5.08 + 3.69i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.591 + 0.814i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.46 + 1.44i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.84 + 5.68i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (4.73 - 6.51i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.13 + 6.57i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.94 - 9.05i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.52 + 6.22i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.26 - 0.922i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 9.94iT - 43T^{2} \)
47 \( 1 + (-4.60 + 1.49i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.68 + 0.873i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.30 + 2.39i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.55 + 1.85i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-3.01 - 0.979i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.01 - 6.19i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.216 + 0.297i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.02 - 3.15i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (13.1 + 4.28i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (2.02 + 1.47i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.173 - 0.0564i)T + (78.4 - 57.0i)T^{2} \)
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

−8.952160809048354137255312717002, −8.507952998277873616149391186368, −7.57065826018876271931589096589, −6.96528581741525014319418134351, −5.66039185717572045583202519442, −5.22597317166061893427473363486, −3.87819198612147514873686138384, −2.88336107028274259369813216193, −2.14165901893921645328204072377, −0.28058888739357635943390640975, 1.85320036099457060881917933373, 2.66488655265890231690283245319, 3.96098775970669530129488653612, 4.61328567677544448008270202607, 5.60911438642637387894447854682, 6.70032979076385453792475022598, 7.58233478479341983421771386675, 8.077213730773683627198050185066, 8.980751460281357576852993174820, 9.984950522083517302501911109549

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