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Properties

Label 2-1500-25.11-c1-0-12
Degree $2$
Conductor $1500$
Sign $0.245 + 0.969i$
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  + (0.309 − 0.951i)3-s + 3.78·7-s + (−0.809 − 0.587i)9-s + (0.653 − 0.474i)11-s + (−3.84 − 2.79i)13-s + (−0.355 − 1.09i)17-s + (0.00463 + 0.0142i)19-s + (1.17 − 3.60i)21-s + (5.07 − 3.68i)23-s + (−0.809 + 0.587i)27-s + (1.14 − 3.51i)29-s + (−0.488 − 1.50i)31-s + (−0.249 − 0.768i)33-s + (6.91 + 5.02i)37-s + (−3.84 + 2.79i)39-s + ⋯
L(s)  = 1  + (0.178 − 0.549i)3-s + 1.43·7-s + (−0.269 − 0.195i)9-s + (0.197 − 0.143i)11-s + (−1.06 − 0.774i)13-s + (−0.0861 − 0.264i)17-s + (0.00106 + 0.00327i)19-s + (0.255 − 0.786i)21-s + (1.05 − 0.768i)23-s + (−0.155 + 0.113i)27-s + (0.212 − 0.653i)29-s + (−0.0878 − 0.270i)31-s + (−0.0434 − 0.133i)33-s + (1.13 + 0.825i)37-s + (−0.615 + 0.447i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.245 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.968309695\)
\(L(\frac12)\) \(\approx\) \(1.968309695\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 \)
good7 \( 1 - 3.78T + 7T^{2} \)
11 \( 1 + (-0.653 + 0.474i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.84 + 2.79i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.355 + 1.09i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.00463 - 0.0142i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-5.07 + 3.68i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.14 + 3.51i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.488 + 1.50i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.91 - 5.02i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (9.30 + 6.75i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + (0.162 - 0.500i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.911 - 2.80i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (9.25 + 6.72i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.54 - 1.84i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-4.10 - 12.6i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.51 - 4.67i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.78 + 2.75i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.86 + 8.81i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.439 - 1.35i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-13.0 + 9.46i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.49 + 7.66i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.151252160508916837079794512260, −8.376961566232945370116571526335, −7.72309367251751797728353279835, −7.12052911151554884565661754003, −6.03124996150480653554196804942, −5.07425290811558575264159496722, −4.43825564719006551925552284290, −2.98137096604477636026350884194, −2.08565810295701590429152671984, −0.816754891274118887613970800200, 1.46866332033316382677643230704, 2.53140382883738215035921224756, 3.80879128295655458527388675512, 4.80133072620661983831890248223, 5.11772808067742571870934669365, 6.42776975188291947359327155793, 7.44999295830984500663578519185, 7.993753768278484071142216880281, 9.054808287648867112148684519854, 9.424841002864519569411885258926

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