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Properties

Label 2-1500-15.2-c1-0-28
Degree $2$
Conductor $1500$
Sign $-0.391 + 0.920i$
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  + (0.678 − 1.59i)3-s + (1.30 + 1.30i)7-s + (−2.07 − 2.16i)9-s − 4.00i·11-s + (0.272 − 0.272i)13-s + (2.24 − 2.24i)17-s + 3.67i·19-s + (2.95 − 1.18i)21-s + (−6.44 − 6.44i)23-s + (−4.85 + 1.84i)27-s + 2.72·29-s + 5.94·31-s + (−6.38 − 2.71i)33-s + (−4.14 − 4.14i)37-s + (−0.248 − 0.618i)39-s + ⋯
L(s)  = 1  + (0.391 − 0.920i)3-s + (0.491 + 0.491i)7-s + (−0.693 − 0.720i)9-s − 1.20i·11-s + (0.0754 − 0.0754i)13-s + (0.544 − 0.544i)17-s + 0.842i·19-s + (0.644 − 0.259i)21-s + (−1.34 − 1.34i)23-s + (−0.934 + 0.355i)27-s + 0.505·29-s + 1.06·31-s + (−1.11 − 0.473i)33-s + (−0.681 − 0.681i)37-s + (−0.0398 − 0.0989i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.391 + 0.920i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.391 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.764144122\)
\(L(\frac12)\) \(\approx\) \(1.764144122\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.678 + 1.59i)T \)
5 \( 1 \)
good7 \( 1 + (-1.30 - 1.30i)T + 7iT^{2} \)
11 \( 1 + 4.00iT - 11T^{2} \)
13 \( 1 + (-0.272 + 0.272i)T - 13iT^{2} \)
17 \( 1 + (-2.24 + 2.24i)T - 17iT^{2} \)
19 \( 1 - 3.67iT - 19T^{2} \)
23 \( 1 + (6.44 + 6.44i)T + 23iT^{2} \)
29 \( 1 - 2.72T + 29T^{2} \)
31 \( 1 - 5.94T + 31T^{2} \)
37 \( 1 + (4.14 + 4.14i)T + 37iT^{2} \)
41 \( 1 - 0.0587iT - 41T^{2} \)
43 \( 1 + (-5.58 + 5.58i)T - 43iT^{2} \)
47 \( 1 + (2.30 - 2.30i)T - 47iT^{2} \)
53 \( 1 + (6.59 + 6.59i)T + 53iT^{2} \)
59 \( 1 - 14.9T + 59T^{2} \)
61 \( 1 + 9.03T + 61T^{2} \)
67 \( 1 + (-0.657 - 0.657i)T + 67iT^{2} \)
71 \( 1 - 7.42iT - 71T^{2} \)
73 \( 1 + (-6.86 + 6.86i)T - 73iT^{2} \)
79 \( 1 + 0.110iT - 79T^{2} \)
83 \( 1 + (6.47 + 6.47i)T + 83iT^{2} \)
89 \( 1 - 0.461T + 89T^{2} \)
97 \( 1 + (-10.1 - 10.1i)T + 97iT^{2} \)
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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

−8.937494775829553756187272355530, −8.260951804564067303271271389231, −7.929418224157967818666810002057, −6.74797612588324973482812151942, −6.03340535869937395978592485034, −5.31784594688595319430960358055, −3.94016202008019673595811207962, −2.93386423374119309723057801884, −1.99681555337379426717948642717, −0.67024797902832914234884658795, 1.60837043943449085164601903220, 2.81182804949735943233004158896, 3.95444201698776005124302058132, 4.56136358100377651086350165630, 5.38061202462728832196436935977, 6.48377348117715052334237450557, 7.61079780145778080680111482631, 8.057577202187330839017141158547, 9.065447185002439365675998847837, 9.869232029535504326804915382491

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