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Properties

Label 2-1500-15.2-c1-0-22
Degree $2$
Conductor $1500$
Sign $0.940 + 0.339i$
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  + (−0.587 + 1.62i)3-s + (1.67 + 1.67i)7-s + (−2.30 − 1.91i)9-s − 3.38i·11-s + (0.729 − 0.729i)13-s + (4.39 − 4.39i)17-s − 5.19i·19-s + (−3.71 + 1.74i)21-s + (−3.36 − 3.36i)23-s + (4.47 − 2.63i)27-s − 6.24·29-s + 3.21·31-s + (5.52 + 1.99i)33-s + (−1.02 − 1.02i)37-s + (0.760 + 1.61i)39-s + ⋯
L(s)  = 1  + (−0.339 + 0.940i)3-s + (0.633 + 0.633i)7-s + (−0.769 − 0.638i)9-s − 1.02i·11-s + (0.202 − 0.202i)13-s + (1.06 − 1.06i)17-s − 1.19i·19-s + (−0.810 + 0.381i)21-s + (−0.701 − 0.701i)23-s + (0.861 − 0.507i)27-s − 1.15·29-s + 0.576·31-s + (0.961 + 0.346i)33-s + (−0.167 − 0.167i)37-s + (0.121 + 0.259i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.940 + 0.339i$
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: \((2,\ 1500,\ (\ :1/2),\ 0.940 + 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.409001497\)
\(L(\frac12)\) \(\approx\) \(1.409001497\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.587 - 1.62i)T \)
5 \( 1 \)
good7 \( 1 + (-1.67 - 1.67i)T + 7iT^{2} \)
11 \( 1 + 3.38iT - 11T^{2} \)
13 \( 1 + (-0.729 + 0.729i)T - 13iT^{2} \)
17 \( 1 + (-4.39 + 4.39i)T - 17iT^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + (3.36 + 3.36i)T + 23iT^{2} \)
29 \( 1 + 6.24T + 29T^{2} \)
31 \( 1 - 3.21T + 31T^{2} \)
37 \( 1 + (1.02 + 1.02i)T + 37iT^{2} \)
41 \( 1 - 3.22iT - 41T^{2} \)
43 \( 1 + (-1.41 + 1.41i)T - 43iT^{2} \)
47 \( 1 + (-8.14 + 8.14i)T - 47iT^{2} \)
53 \( 1 + (-1.42 - 1.42i)T + 53iT^{2} \)
59 \( 1 - 3.16T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 + (-11.1 - 11.1i)T + 67iT^{2} \)
71 \( 1 + 16.4iT - 71T^{2} \)
73 \( 1 + (9.57 - 9.57i)T - 73iT^{2} \)
79 \( 1 - 13.1iT - 79T^{2} \)
83 \( 1 + (1.91 + 1.91i)T + 83iT^{2} \)
89 \( 1 - 3.69T + 89T^{2} \)
97 \( 1 + (-1.81 - 1.81i)T + 97iT^{2} \)
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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

−9.387736727352932085655266477234, −8.753836608157725456797417936985, −8.077198882252108995932376381342, −6.95336824615932966546376483042, −5.78975042926940756183103093809, −5.40049251698446755632462592669, −4.47585382751044941283641780159, −3.42294313696431904617312167069, −2.51602088492400416287205904052, −0.63881029627763997073606114939, 1.31023461742397699109864820852, 1.98352185418295507974818779026, 3.56426763536678312054137572899, 4.49727545002231846978735548171, 5.62952277193619788480124455355, 6.19009685316296816749937506984, 7.43447651201146523735650891840, 7.63533358838004671155663541393, 8.448645405260986462536200071116, 9.629009503772096388762441016037

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