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Properties

Label 2-1500-25.9-c1-0-3
Degree $2$
Conductor $1500$
Sign $0.325 - 0.945i$
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 + 1.50i·7-s + (0.809 − 0.587i)9-s + (4.99 + 3.62i)11-s + (−2.09 − 2.87i)13-s + (0.471 + 0.153i)17-s + (−0.0963 + 0.296i)19-s + (−0.464 − 1.43i)21-s + (1.79 − 2.47i)23-s + (−0.587 + 0.809i)27-s + (0.0378 + 0.116i)29-s + (−0.909 + 2.79i)31-s + (−5.87 − 1.90i)33-s + (2.56 + 3.53i)37-s + (2.87 + 2.09i)39-s + ⋯
L(s)  = 1  + (−0.549 + 0.178i)3-s + 0.568i·7-s + (0.269 − 0.195i)9-s + (1.50 + 1.09i)11-s + (−0.579 − 0.797i)13-s + (0.114 + 0.0371i)17-s + (−0.0220 + 0.0680i)19-s + (−0.101 − 0.312i)21-s + (0.375 − 0.516i)23-s + (−0.113 + 0.155i)27-s + (0.00701 + 0.0216i)29-s + (−0.163 + 0.502i)31-s + (−1.02 − 0.332i)33-s + (0.421 + 0.580i)37-s + (0.460 + 0.334i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.325 - 0.945i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.325 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.362350634\)
\(L(\frac12)\) \(\approx\) \(1.362350634\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.50iT - 7T^{2} \)
11 \( 1 + (-4.99 - 3.62i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.09 + 2.87i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.471 - 0.153i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.0963 - 0.296i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.79 + 2.47i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.0378 - 0.116i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.909 - 2.79i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.56 - 3.53i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.44 - 2.50i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.62iT - 43T^{2} \)
47 \( 1 + (5.02 - 1.63i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (8.17 - 2.65i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-10.4 + 7.57i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-9.15 - 6.65i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-12.5 - 4.09i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.00 - 3.10i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (9.38 - 12.9i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.63 - 8.11i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (10.7 + 3.50i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-11.4 - 8.30i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-10.9 + 3.54i)T + (78.4 - 57.0i)T^{2} \)
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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

−9.763985039704070740259555910464, −8.958519563059543278827796288235, −8.074823256127552587143323890102, −6.99781177714905507309592560986, −6.48493967268864172877580915487, −5.43981839902351401799268895309, −4.71489985670199054181835538022, −3.78193736930425394556797754597, −2.53081633459805012634376074482, −1.22879827767855654385783666959, 0.67090103300954349375175264491, 1.87287024447471654295577369256, 3.45505639259026150890515087717, 4.18510482213983580967170226571, 5.22306593968669199813705658688, 6.18585229934598258744933332108, 6.81855179559749303208930051973, 7.53767379403188275939726334037, 8.634734917379190452123726116108, 9.323962117093687014518094628014

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