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Properties

Label 2-750-125.104-c1-0-20
Degree $2$
Conductor $750$
Sign $0.997 - 0.0734i$
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 + (1.41 − 1.73i)5-s + (0.425 + 0.904i)6-s + (0.223 − 0.307i)7-s + (−0.684 − 0.728i)8-s + (0.929 − 0.368i)9-s + (2.03 + 0.936i)10-s + (2.80 − 0.720i)11-s + (−0.770 + 0.637i)12-s + (−0.816 − 2.06i)13-s + (0.353 + 0.139i)14-s + (1.06 − 1.96i)15-s + (0.535 − 0.844i)16-s + (−0.785 + 1.42i)17-s + ⋯
L(s)  = 1  + (0.175 + 0.684i)2-s + (0.567 − 0.108i)3-s + (−0.438 + 0.240i)4-s + (0.631 − 0.775i)5-s + (0.173 + 0.369i)6-s + (0.0844 − 0.116i)7-s + (−0.242 − 0.257i)8-s + (0.309 − 0.122i)9-s + (0.642 + 0.296i)10-s + (0.845 − 0.217i)11-s + (−0.222 + 0.184i)12-s + (−0.226 − 0.572i)13-s + (0.0944 + 0.0374i)14-s + (0.274 − 0.508i)15-s + (0.133 − 0.211i)16-s + (−0.190 + 0.346i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.997 - 0.0734i$
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: \((2,\ 750,\ (\ :1/2),\ 0.997 - 0.0734i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21766 + 0.0815935i\)
\(L(\frac12)\) \(\approx\) \(2.21766 + 0.0815935i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 1.73i)T \)
good7 \( 1 + (-0.223 + 0.307i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (-2.80 + 0.720i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (0.816 + 2.06i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (0.785 - 1.42i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (-0.556 + 2.91i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (-0.243 - 0.0153i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (-0.828 - 0.104i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (-3.33 - 1.83i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (-5.21 - 3.30i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (-0.0480 - 0.764i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (-1.28 + 0.416i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (1.04 - 1.11i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (5.83 + 2.74i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (6.54 + 7.91i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (0.475 - 7.56i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (-0.714 - 5.65i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (-4.00 - 3.76i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (-3.36 - 2.78i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (-0.886 - 4.64i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (12.4 + 2.37i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (4.97 - 6.01i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (1.91 - 15.1i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.00325754721550647969664067076, −9.331613944060158800138923291867, −8.586259078616442133774651284102, −7.907940098013960434232432782596, −6.80186001539539858188162667004, −6.01603124752829541251152889190, −4.96661648606153845783969462248, −4.11111327259962400265726840236, −2.76750506896583844179607956334, −1.18118539816576684344545464092, 1.65949202038179219885029591350, 2.62134813746724934783404874757, 3.67036126472002536516017468698, 4.65348934443695921800380218439, 5.92598301206928422586985818075, 6.79607964157036721018462131233, 7.81138915127042133062337638372, 9.007689487476541018386092741720, 9.557721804312858552063491487710, 10.25609100976128437578947098049

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