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Properties

Label 2-1500-25.6-c1-0-1
Degree $2$
Conductor $1500$
Sign $-0.857 - 0.514i$
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  + (−0.809 + 0.587i)3-s − 0.957·7-s + (0.309 − 0.951i)9-s + (−1.67 − 5.15i)11-s + (−0.625 + 1.92i)13-s + (0.520 + 0.377i)17-s + (4.07 + 2.96i)19-s + (0.774 − 0.562i)21-s + (1.08 + 3.34i)23-s + (0.309 + 0.951i)27-s + (−8.20 + 5.96i)29-s + (−2.98 − 2.16i)31-s + (4.38 + 3.18i)33-s + (−3.49 + 10.7i)37-s + (−0.625 − 1.92i)39-s + ⋯
L(s)  = 1  + (−0.467 + 0.339i)3-s − 0.361·7-s + (0.103 − 0.317i)9-s + (−0.504 − 1.55i)11-s + (−0.173 + 0.533i)13-s + (0.126 + 0.0916i)17-s + (0.935 + 0.679i)19-s + (0.169 − 0.122i)21-s + (0.226 + 0.697i)23-s + (0.0594 + 0.183i)27-s + (−1.52 + 1.10i)29-s + (−0.536 − 0.389i)31-s + (0.762 + 0.554i)33-s + (−0.574 + 1.76i)37-s + (−0.100 − 0.308i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.857 - 0.514i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.857 - 0.514i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4302675224\)
\(L(\frac12)\) \(\approx\) \(0.4302675224\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 \)
good7 \( 1 + 0.957T + 7T^{2} \)
11 \( 1 + (1.67 + 5.15i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.625 - 1.92i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.520 - 0.377i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.07 - 2.96i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.08 - 3.34i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (8.20 - 5.96i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.98 + 2.16i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.49 - 10.7i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.08 + 3.35i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.766T + 43T^{2} \)
47 \( 1 + (3.99 - 2.90i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.81 - 3.49i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.45 + 4.48i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.34 - 4.13i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (7.70 + 5.59i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-9.66 + 7.02i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.67 + 5.16i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.58 - 6.96i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.12 - 0.819i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.527 + 1.62i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (11.8 - 8.57i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.725556978719075856347848100973, −9.188839517336475797249820693521, −8.223909939183895785386951266060, −7.41178164887361641925273127629, −6.41949702268944027828747611940, −5.64544296340787664662333850034, −5.03628148127940228546973214354, −3.67221180731689572675608515284, −3.13443671131468077483479968740, −1.43233320553070820463044784456, 0.18101579416797896244670444522, 1.83291242132159622885641425512, 2.86494564560478375255836038491, 4.15628982455890005470580320375, 5.11672050762529532756267488948, 5.74590586675971014548402384105, 6.97479276732170930365974929318, 7.30980509705776739098390015336, 8.187373212582602492336994884193, 9.423347400688169339679428843428

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