Properties

Label 2-1500-25.14-c1-0-11
Degree $2$
Conductor $1500$
Sign $-0.609 + 0.792i$
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 − 3.54i·7-s + (0.809 + 0.587i)9-s + (1.78 − 1.29i)11-s + (4.21 − 5.80i)13-s + (−6.05 + 1.96i)17-s + (0.715 + 2.20i)19-s + (−1.09 + 3.37i)21-s + (1.27 + 1.76i)23-s + (−0.587 − 0.809i)27-s + (−0.262 + 0.806i)29-s + (−1.32 − 4.09i)31-s + (−2.09 + 0.681i)33-s + (−4.24 + 5.84i)37-s + (−5.80 + 4.21i)39-s + ⋯
L(s)  = 1  + (−0.549 − 0.178i)3-s − 1.34i·7-s + (0.269 + 0.195i)9-s + (0.538 − 0.390i)11-s + (1.17 − 1.61i)13-s + (−1.46 + 0.477i)17-s + (0.164 + 0.504i)19-s + (−0.239 + 0.736i)21-s + (0.266 + 0.367i)23-s + (−0.113 − 0.155i)27-s + (−0.0486 + 0.149i)29-s + (−0.238 − 0.734i)31-s + (−0.365 + 0.118i)33-s + (−0.698 + 0.961i)37-s + (−0.929 + 0.675i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.609 + 0.792i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.609 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.112784183\)
\(L(\frac12)\) \(\approx\) \(1.112784183\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.54iT - 7T^{2} \)
11 \( 1 + (-1.78 + 1.29i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-4.21 + 5.80i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (6.05 - 1.96i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.715 - 2.20i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.27 - 1.76i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.262 - 0.806i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.32 + 4.09i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.24 - 5.84i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.08 - 0.790i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.18iT - 43T^{2} \)
47 \( 1 + (-5.75 - 1.87i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (11.3 + 3.69i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (10.0 + 7.33i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-5.59 + 4.06i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.50 - 1.46i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-4.25 + 13.1i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.640 - 0.881i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.80 + 5.55i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-11.9 + 3.87i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (5.68 - 4.12i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (17.2 + 5.60i)T + (78.4 + 57.0i)T^{2} \)
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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

−9.200597822368780842105947517007, −8.247974180185240964354663993768, −7.61914838329022428477874626222, −6.62427256314160960865397733952, −6.08599297986177029814298550250, −5.06297182609013237148690077939, −3.99678593432133059876215843416, −3.34533851865366383465839439014, −1.57855438301821969377070977227, −0.49425637878122053187265033901, 1.58152548293168791738856076083, 2.61782743769404532546997612445, 4.04078210885349352463596929454, 4.72172752671793763771249170162, 5.73479126283810579267357822505, 6.53026882087365534694606260211, 7.01341864688823251873445579051, 8.473815325900916640773495346413, 9.203808028679798227133408454090, 9.318518287964965859608018889803

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