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Properties

Label 2-1500-25.16-c1-0-13
Degree $2$
Conductor $1500$
Sign $0.387 + 0.921i$
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 + 4.78·7-s + (−0.809 + 0.587i)9-s + (−1.58 − 1.14i)11-s + (0.873 − 0.634i)13-s + (1.17 − 3.61i)17-s + (1.31 − 4.04i)19-s + (−1.47 − 4.54i)21-s + (−4.74 − 3.44i)23-s + (0.809 + 0.587i)27-s + (3.26 + 10.0i)29-s + (−1.33 + 4.10i)31-s + (−0.604 + 1.86i)33-s + (4.57 − 3.32i)37-s + (−0.873 − 0.634i)39-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + 1.80·7-s + (−0.269 + 0.195i)9-s + (−0.477 − 0.346i)11-s + (0.242 − 0.176i)13-s + (0.285 − 0.877i)17-s + (0.301 − 0.927i)19-s + (−0.322 − 0.992i)21-s + (−0.989 − 0.718i)23-s + (0.155 + 0.113i)27-s + (0.605 + 1.86i)29-s + (−0.239 + 0.737i)31-s + (−0.105 + 0.323i)33-s + (0.752 − 0.546i)37-s + (−0.139 − 0.101i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.387 + 0.921i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.387 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.876051331\)
\(L(\frac12)\) \(\approx\) \(1.876051331\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.78T + 7T^{2} \)
11 \( 1 + (1.58 + 1.14i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.873 + 0.634i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.17 + 3.61i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.31 + 4.04i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (4.74 + 3.44i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-3.26 - 10.0i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.33 - 4.10i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.57 + 3.32i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.694 - 0.504i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + (0.927 + 2.85i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.30 + 4.01i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.85 + 2.80i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.93 + 2.13i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.14 - 6.59i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (3.70 + 11.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (13.7 + 9.96i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.04 + 6.29i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.797 + 2.45i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-0.673 - 0.489i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.81 - 8.67i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.039603846653385392237043141033, −8.482070758489021088669412782593, −7.66757864925517254212446426685, −7.15402764002254926041208782945, −5.96417637907092910451774964104, −5.11679627678389325538623555045, −4.56479386093055728956203047343, −3.05866839881666604064673477105, −1.98976290860013901943078055657, −0.855504948465429971280967725636, 1.37192937843750562103290648975, 2.42942909431843572331434172508, 4.04097324952493048715829613651, 4.41796580345496336991799889841, 5.57591386961440927686072996849, 6.01573685062692472006608928860, 7.65655602703002912917222703726, 7.897331593846316022119394408209, 8.724486169824018417996319151092, 9.835994000471904326202003312275

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