Properties

Label 2-1500-25.9-c1-0-10
Degree $2$
Conductor $1500$
Sign $0.0796 + 0.996i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 − 0.309i)3-s + 1.74i·7-s + (0.809 − 0.587i)9-s + (−1.87 − 1.36i)11-s + (−2.90 − 3.99i)13-s + (−3.55 − 1.15i)17-s + (0.523 − 1.61i)19-s + (0.537 + 1.65i)21-s + (5.10 − 7.02i)23-s + (0.587 − 0.809i)27-s + (−0.964 − 2.96i)29-s + (2.95 − 9.09i)31-s + (−2.20 − 0.717i)33-s + (3.15 + 4.34i)37-s + (−3.99 − 2.90i)39-s + ⋯
L(s)  = 1  + (0.549 − 0.178i)3-s + 0.657i·7-s + (0.269 − 0.195i)9-s + (−0.565 − 0.411i)11-s + (−0.805 − 1.10i)13-s + (−0.862 − 0.280i)17-s + (0.120 − 0.369i)19-s + (0.117 + 0.361i)21-s + (1.06 − 1.46i)23-s + (0.113 − 0.155i)27-s + (−0.179 − 0.551i)29-s + (0.530 − 1.63i)31-s + (−0.384 − 0.124i)33-s + (0.518 + 0.713i)37-s + (−0.639 − 0.464i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0796 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0796 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.0796 + 0.996i$
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: \((2,\ 1500,\ (\ :1/2),\ 0.0796 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.575469994\)
\(L(\frac12)\) \(\approx\) \(1.575469994\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.74iT - 7T^{2} \)
11 \( 1 + (1.87 + 1.36i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.90 + 3.99i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.55 + 1.15i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.523 + 1.61i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-5.10 + 7.02i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.964 + 2.96i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.95 + 9.09i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.15 - 4.34i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.05 + 5.12i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.86iT - 43T^{2} \)
47 \( 1 + (8.04 - 2.61i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.27 + 0.415i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.54 + 2.57i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-12.4 - 9.01i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (8.77 + 2.85i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.00728 - 0.0224i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.601 - 0.827i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.246 - 0.759i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.25 - 0.732i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (3.93 + 2.85i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.29 - 1.06i)T + (78.4 - 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.229188767823058571663288614536, −8.430634965240426847741418448760, −7.85237402369333892709393951533, −6.95394275553667120042765087650, −6.01269113007277056871779095074, −5.12922535234505135985920031465, −4.23768217593333879729785975951, −2.66584786698570245451464840298, −2.59722607352814864175542527617, −0.57454193732095069284664111586, 1.52000931281250987202240748550, 2.62749884884871147570755598201, 3.72591136318512672747608366086, 4.57759352561281848638856817102, 5.34073675098151836423542949738, 6.77149316084566064323213587377, 7.20008150297737874993828228050, 8.044153014499063784051691198280, 9.013839873030921516151784274688, 9.567175209251153193497334029920

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