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Properties

Label 2-1500-25.19-c1-0-8
Degree $2$
Conductor $1500$
Sign $0.926 + 0.376i$
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 − 0.809i)3-s + 0.511i·7-s + (−0.309 + 0.951i)9-s + (−0.564 − 1.73i)11-s + (5.82 + 1.89i)13-s + (−1.51 + 2.09i)17-s + (−3.93 − 2.85i)19-s + (0.413 − 0.300i)21-s + (6.30 − 2.04i)23-s + (0.951 − 0.309i)27-s + (−6.46 + 4.70i)29-s + (3.95 + 2.87i)31-s + (−1.07 + 1.47i)33-s + (7.07 + 2.29i)37-s + (−1.89 − 5.82i)39-s + ⋯
L(s)  = 1  + (−0.339 − 0.467i)3-s + 0.193i·7-s + (−0.103 + 0.317i)9-s + (−0.170 − 0.523i)11-s + (1.61 + 0.524i)13-s + (−0.368 + 0.506i)17-s + (−0.902 − 0.655i)19-s + (0.0902 − 0.0655i)21-s + (1.31 − 0.427i)23-s + (0.183 − 0.0594i)27-s + (−1.20 + 0.872i)29-s + (0.709 + 0.515i)31-s + (−0.186 + 0.257i)33-s + (1.16 + 0.377i)37-s + (−0.302 − 0.932i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.926 + 0.376i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.926 + 0.376i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.538913335\)
\(L(\frac12)\) \(\approx\) \(1.538913335\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 \)
good7 \( 1 - 0.511iT - 7T^{2} \)
11 \( 1 + (0.564 + 1.73i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-5.82 - 1.89i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.51 - 2.09i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (3.93 + 2.85i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-6.30 + 2.04i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (6.46 - 4.70i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.95 - 2.87i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-7.07 - 2.29i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.77 + 5.45i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.05iT - 43T^{2} \)
47 \( 1 + (-4.67 - 6.43i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.781 + 1.07i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.11 + 9.59i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.47 - 4.53i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-8.42 + 11.5i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-4.34 + 3.15i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.31 + 1.07i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.06 - 0.775i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.536 + 0.738i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.63 + 11.1i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.34 + 5.98i)T + (-29.9 + 92.2i)T^{2} \)
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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

−9.055353444278674487317993209082, −8.790429938598189538173052543555, −7.87939505455791956628810044645, −6.78211125342691829124466798762, −6.30759143402606344904352731948, −5.43655693432334823029542872954, −4.39179195307830698248816609102, −3.37010078085915147884199306546, −2.15117483101432310916879554661, −0.908894174669201284789626459448, 0.944252414044680485976841941114, 2.47274512540278457728979872976, 3.73156038410379482866773842551, 4.36087296245114897545086421059, 5.49652341017701373012336849086, 6.12436182562969143333513385808, 7.07134423950888590690935363512, 8.003074455809343352351662612868, 8.784430843340926026405478978622, 9.598521353844057511387843488339

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