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Properties

Label 2-1500-15.2-c1-0-29
Degree $2$
Conductor $1500$
Sign $-0.780 + 0.625i$
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  + (0.190 − 1.72i)3-s + (−0.111 − 0.111i)7-s + (−2.92 − 0.654i)9-s + 1.01i·11-s + (2.98 − 2.98i)13-s + (1.27 − 1.27i)17-s − 4.73i·19-s + (−0.213 + 0.170i)21-s + (0.327 + 0.327i)23-s + (−1.68 + 4.91i)27-s − 6.26·29-s − 3.35·31-s + (1.74 + 0.192i)33-s + (−1.73 − 1.73i)37-s + (−4.56 − 5.70i)39-s + ⋯
L(s)  = 1  + (0.109 − 0.993i)3-s + (−0.0421 − 0.0421i)7-s + (−0.975 − 0.218i)9-s + 0.305i·11-s + (0.826 − 0.826i)13-s + (0.308 − 0.308i)17-s − 1.08i·19-s + (−0.0465 + 0.0372i)21-s + (0.0682 + 0.0682i)23-s + (−0.324 + 0.946i)27-s − 1.16·29-s − 0.602·31-s + (0.303 + 0.0335i)33-s + (−0.284 − 0.284i)37-s + (−0.731 − 0.912i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.780 + 0.625i$
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: \((2,\ 1500,\ (\ :1/2),\ -0.780 + 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.312045988\)
\(L(\frac12)\) \(\approx\) \(1.312045988\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.190 + 1.72i)T \)
5 \( 1 \)
good7 \( 1 + (0.111 + 0.111i)T + 7iT^{2} \)
11 \( 1 - 1.01iT - 11T^{2} \)
13 \( 1 + (-2.98 + 2.98i)T - 13iT^{2} \)
17 \( 1 + (-1.27 + 1.27i)T - 17iT^{2} \)
19 \( 1 + 4.73iT - 19T^{2} \)
23 \( 1 + (-0.327 - 0.327i)T + 23iT^{2} \)
29 \( 1 + 6.26T + 29T^{2} \)
31 \( 1 + 3.35T + 31T^{2} \)
37 \( 1 + (1.73 + 1.73i)T + 37iT^{2} \)
41 \( 1 + 5.46iT - 41T^{2} \)
43 \( 1 + (-3.13 + 3.13i)T - 43iT^{2} \)
47 \( 1 + (-6.07 + 6.07i)T - 47iT^{2} \)
53 \( 1 + (4.90 + 4.90i)T + 53iT^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 - 2.26T + 61T^{2} \)
67 \( 1 + (-5.84 - 5.84i)T + 67iT^{2} \)
71 \( 1 - 13.4iT - 71T^{2} \)
73 \( 1 + (6.11 - 6.11i)T - 73iT^{2} \)
79 \( 1 + 6.81iT - 79T^{2} \)
83 \( 1 + (-3.18 - 3.18i)T + 83iT^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 + (7.71 + 7.71i)T + 97iT^{2} \)
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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

−8.946197282637834424909578398206, −8.394255866963552891595727300114, −7.35959915460023702240442797767, −7.00638694659618434906224177623, −5.84971412182365595703788124223, −5.29901813738304212118646774119, −3.86420393097370954721980728607, −2.90930093205815418573776672564, −1.81956581161196119853238684619, −0.50949878383955910006917818607, 1.62470902394163715040174518304, 3.07537058520841286043191478124, 3.85810052708860113124337006020, 4.63156114607466740287092099033, 5.78504157679581990198948414309, 6.22568939542700894355749525003, 7.60171613184128615814587107867, 8.297469272664959512801951862270, 9.232695390534852123112747192373, 9.562509402839941217755672660058

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