↑

Properties

Label 2-750-25.6-c1-0-11
Degree $2$
Conductor $750$
Sign $0.432 + 0.901i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s − 0.329·7-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (−1.55 − 4.78i)11-s + (0.309 − 0.951i)12-s + (−0.148 + 0.458i)13-s + (−0.101 − 0.313i)14-s + (0.309 − 0.951i)16-s + (−5.49 − 3.98i)17-s + 18-s + (−4.40 − 3.19i)19-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.330 − 0.239i)6-s − 0.124·7-s + (−0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + (−0.468 − 1.44i)11-s + (0.0892 − 0.274i)12-s + (−0.0413 + 0.127i)13-s + (−0.0271 − 0.0837i)14-s + (0.0772 − 0.237i)16-s + (−1.33 − 0.967i)17-s + 0.235·18-s + (−1.00 − 0.733i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.432 + 0.901i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.432 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.540758 - 0.340495i\)
\(L(\frac12)\) \(\approx\) \(0.540758 - 0.340495i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 \)
good7 \( 1 + 0.329T + 7T^{2} \)
11 \( 1 + (1.55 + 4.78i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.148 - 0.458i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (5.49 + 3.98i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.40 + 3.19i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.00 - 6.18i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.87 + 3.53i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.06 + 0.775i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.241 + 0.741i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.86 + 11.9i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.47T + 43T^{2} \)
47 \( 1 + (-3.54 + 2.57i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.36 + 0.990i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.313 + 0.966i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.29 - 3.98i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (2.54 + 1.84i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (4.62 - 3.35i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.909 - 2.79i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.86 - 4.98i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (13.9 + 10.1i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.06 - 3.28i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-7.69 + 5.58i)T + (29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.23644045737676754909477240678, −9.074551991413221428646534771412, −8.644277195209368627692009553765, −7.42562515648998665672230152867, −6.58770605362732776688092664668, −5.75634335992465227185487688802, −4.92052970467497234054536367023, −3.92870600118873390211056345863, −2.69319120662854101075978637911, −0.31285239038542887254119292087, 1.66864292387223894586382988635, 2.67107963208851466272761067391, 4.29354554232069085308915737549, 4.80430779792861705392841547473, 6.16836219267839531753838807325, 6.80684879331855663881159492075, 8.018529431078495286164502516898, 8.843077173112177504915971122993, 10.01655259880259985178616709415, 10.53393269006657033662019176226

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