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Properties

Label 2-750-25.14-c1-0-6
Degree $2$
Conductor $750$
Sign $-0.357 - 0.934i$
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  + (0.587 + 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.309 + 0.951i)6-s + 2i·7-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (0.618 − 0.449i)11-s + (−0.587 + 0.809i)12-s + (−1.08 + 1.5i)13-s + (−1.61 + 1.17i)14-s + (−0.809 − 0.587i)16-s + (−1.08 + 0.354i)17-s + 0.999i·18-s + (2.23 + 6.88i)19-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (0.549 + 0.178i)3-s + (−0.154 + 0.475i)4-s + (0.126 + 0.388i)6-s + 0.755i·7-s + (−0.336 + 0.109i)8-s + (0.269 + 0.195i)9-s + (0.186 − 0.135i)11-s + (−0.169 + 0.233i)12-s + (−0.302 + 0.416i)13-s + (−0.432 + 0.314i)14-s + (−0.202 − 0.146i)16-s + (−0.264 + 0.0858i)17-s + 0.235i·18-s + (0.512 + 1.57i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.357 - 0.934i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.357 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20727 + 1.75388i\)
\(L(\frac12)\) \(\approx\) \(1.20727 + 1.75388i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + (-0.618 + 0.449i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.08 - 1.5i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.08 - 0.354i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.23 - 6.88i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.52 - 4.85i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.11 + 3.44i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3 + 9.23i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.20 - 7.16i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.11 + 2.99i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.23iT - 43T^{2} \)
47 \( 1 + (-8.78 - 2.85i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-10.9 - 3.57i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.23 + 5.25i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.73 + 1.26i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.52 - 1.14i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.52 + 7.77i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.79 + 7.97i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-5.70 + 1.85i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-2.92 + 2.12i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (6.79 + 2.20i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.49135215071802266355583657740, −9.492256662104595265109371363933, −8.897587302148352163877788468501, −7.964251463147780575755887671202, −7.24328063582922978542562162973, −6.07538543737048617229035131118, −5.36214601675700673729085975629, −4.18682724529873979598448462326, −3.27173210398008600818189918657, −1.98709266128741669668506756108, 0.939225477047680028055061863041, 2.46620059437453519745052095663, 3.39836402484890621659801127265, 4.49158277362265575804894885813, 5.32580517365088127494133903100, 6.88109196217028189924620526212, 7.21287474006838127837088137649, 8.659823034035812435320970674530, 9.166372482298006587342455048917, 10.36403921702589439991262202228

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