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Properties

Label 2-750-125.104-c1-0-10
Degree $2$
Conductor $750$
Sign $0.999 + 0.00272i$
Analytic cond. $5.98878$
Root an. cond. $2.44719$
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  + (−0.248 − 0.968i)2-s + (0.982 − 0.187i)3-s + (−0.876 + 0.481i)4-s + (2.19 + 0.411i)5-s + (−0.425 − 0.904i)6-s + (−2.79 + 3.84i)7-s + (0.684 + 0.728i)8-s + (0.929 − 0.368i)9-s + (−0.147 − 2.23i)10-s + (3.56 − 0.914i)11-s + (−0.770 + 0.637i)12-s + (−1.15 − 2.90i)13-s + (4.42 + 1.75i)14-s + (2.23 − 0.00733i)15-s + (0.535 − 0.844i)16-s + (−1.83 + 3.34i)17-s + ⋯
L(s)  = 1  + (−0.175 − 0.684i)2-s + (0.567 − 0.108i)3-s + (−0.438 + 0.240i)4-s + (0.982 + 0.184i)5-s + (−0.173 − 0.369i)6-s + (−1.05 + 1.45i)7-s + (0.242 + 0.257i)8-s + (0.309 − 0.122i)9-s + (−0.0467 − 0.705i)10-s + (1.07 − 0.275i)11-s + (−0.222 + 0.184i)12-s + (−0.319 − 0.806i)13-s + (1.18 + 0.467i)14-s + (0.577 − 0.00189i)15-s + (0.133 − 0.211i)16-s + (−0.445 + 0.810i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.999 + 0.00272i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.999 + 0.00272i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78563 - 0.00243581i\)
\(L(\frac12)\) \(\approx\) \(1.78563 - 0.00243581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.248 + 0.968i)T \)
3 \( 1 + (-0.982 + 0.187i)T \)
5 \( 1 + (-2.19 - 0.411i)T \)
good7 \( 1 + (2.79 - 3.84i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (-3.56 + 0.914i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (1.15 + 2.90i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (1.83 - 3.34i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (0.583 - 3.05i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (-4.96 - 0.312i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (-4.66 - 0.589i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (-5.58 - 3.07i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (-4.11 - 2.61i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (-0.418 - 6.65i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (1.46 - 0.476i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (1.04 - 1.11i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (11.9 + 5.60i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (2.87 + 3.47i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (-0.802 + 12.7i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (-1.00 - 7.98i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (6.88 + 6.46i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (3.57 + 2.95i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (-1.55 - 8.17i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (-9.17 - 1.75i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-7.43 + 8.99i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (-1.75 + 13.9i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.03415532915475532704474081818, −9.596713627704012580607153494273, −8.839071465118446845593453790219, −8.218123666558952215150530610179, −6.54978081059003109292066456045, −6.13463822960374353718283445486, −4.89609717336177028865848689131, −3.29329932335104064196810143908, −2.75756264928262653473775588105, −1.55620248397102329937018924425, 1.02612806672545578562361323684, 2.71730916234355617962579849651, 4.10598430851284610556871170108, 4.79197158012095321561911081856, 6.41402430362536510394965841889, 6.74910536260549950943666621141, 7.53398269894382056749709817451, 9.043867806797056406094882018007, 9.277611247224934363593279341920, 10.02247373874866511355926501000

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