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Properties

Label 2-1500-25.11-c1-0-1
Degree $2$
Conductor $1500$
Sign $-0.605 - 0.795i$
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 + 0.595·7-s + (−0.809 − 0.587i)9-s + (−2.71 + 1.97i)11-s + (3.85 + 2.80i)13-s + (−2.31 − 7.11i)17-s + (1.91 + 5.88i)19-s + (−0.184 + 0.566i)21-s + (−3.56 + 2.59i)23-s + (0.809 − 0.587i)27-s + (−0.853 + 2.62i)29-s + (1.38 + 4.26i)31-s + (−1.03 − 3.19i)33-s + (−1.05 − 0.764i)37-s + (−3.85 + 2.80i)39-s + ⋯
L(s)  = 1  + (−0.178 + 0.549i)3-s + 0.225·7-s + (−0.269 − 0.195i)9-s + (−0.818 + 0.594i)11-s + (1.07 + 0.777i)13-s + (−0.560 − 1.72i)17-s + (0.438 + 1.35i)19-s + (−0.0401 + 0.123i)21-s + (−0.743 + 0.540i)23-s + (0.155 − 0.113i)27-s + (−0.158 + 0.487i)29-s + (0.249 + 0.766i)31-s + (−0.180 − 0.555i)33-s + (−0.172 − 0.125i)37-s + (−0.617 + 0.448i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.605 - 0.795i$
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: \((2,\ 1500,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.102154386\)
\(L(\frac12)\) \(\approx\) \(1.102154386\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.595T + 7T^{2} \)
11 \( 1 + (2.71 - 1.97i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.85 - 2.80i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.31 + 7.11i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.91 - 5.88i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.56 - 2.59i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.853 - 2.62i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.38 - 4.26i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.05 + 0.764i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.61 - 5.53i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.59T + 43T^{2} \)
47 \( 1 + (-1.36 + 4.18i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.53 - 7.80i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.80 - 1.31i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (10.2 - 7.41i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.58 - 7.94i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.09 + 6.46i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.91 + 4.29i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.26 - 10.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.29 - 3.97i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (3.43 - 2.49i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.79 + 11.6i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.740997603617651146987166036713, −9.115385855759416324996012701539, −8.184892844941887870720413349470, −7.40433441776736357659906077725, −6.47634299721943739033298050245, −5.52102380552126245035625071819, −4.77967521267525086619257359597, −3.90138617946888917977997086398, −2.84202249392209014118016937478, −1.52188767494737203311421618119, 0.44630629217350749735248173992, 1.86242244234049873481839652495, 2.98519701463242940346589431230, 4.05850808281542909756133368393, 5.19378285538763428032444361421, 6.03966552032448611145719536488, 6.57778227562758679567492304331, 7.912415224907025586354719991892, 8.143443821192019853102600968103, 9.011146257750559095080703776489

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