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Properties

Label 2-750-75.47-c1-0-11
Degree $2$
Conductor $750$
Sign $-0.855 - 0.518i$
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  + (0.453 + 0.891i)2-s + (0.333 + 1.69i)3-s + (−0.587 + 0.809i)4-s + (−1.36 + 1.06i)6-s + (2.58 + 2.58i)7-s + (−0.987 − 0.156i)8-s + (−2.77 + 1.13i)9-s + (1.45 + 0.473i)11-s + (−1.57 − 0.729i)12-s + (4.48 + 2.28i)13-s + (−1.12 + 3.47i)14-s + (−0.309 − 0.951i)16-s + (0.806 − 5.09i)17-s + (−2.27 − 1.95i)18-s + (1.27 + 1.75i)19-s + ⋯
L(s)  = 1  + (0.321 + 0.630i)2-s + (0.192 + 0.981i)3-s + (−0.293 + 0.404i)4-s + (−0.556 + 0.436i)6-s + (0.976 + 0.976i)7-s + (−0.349 − 0.0553i)8-s + (−0.925 + 0.378i)9-s + (0.439 + 0.142i)11-s + (−0.453 − 0.210i)12-s + (1.24 + 0.633i)13-s + (−0.301 + 0.928i)14-s + (−0.0772 − 0.237i)16-s + (0.195 − 1.23i)17-s + (−0.535 − 0.461i)18-s + (0.292 + 0.402i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.855 - 0.518i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.855 - 0.518i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.542174 + 1.93926i\)
\(L(\frac12)\) \(\approx\) \(0.542174 + 1.93926i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.453 - 0.891i)T \)
3 \( 1 + (-0.333 - 1.69i)T \)
5 \( 1 \)
good7 \( 1 + (-2.58 - 2.58i)T + 7iT^{2} \)
11 \( 1 + (-1.45 - 0.473i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (-4.48 - 2.28i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.806 + 5.09i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (-1.27 - 1.75i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (5.66 - 2.88i)T + (13.5 - 18.6i)T^{2} \)
29 \( 1 + (5.64 + 4.10i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (5.95 - 4.32i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.48 + 6.84i)T + (-21.7 - 29.9i)T^{2} \)
41 \( 1 + (-2.85 + 0.929i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-3.24 + 3.24i)T - 43iT^{2} \)
47 \( 1 + (-2.81 + 0.446i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (-0.161 - 1.02i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-2.10 - 6.48i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.78 + 5.49i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (3.33 + 0.527i)T + (63.7 + 20.7i)T^{2} \)
71 \( 1 + (2.18 - 3.00i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.270 - 0.531i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (-0.782 + 1.07i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.73 - 0.432i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 + (-1.98 + 6.12i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-2.14 - 13.5i)T + (-92.2 + 29.9i)T^{2} \)
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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

−10.81193884478891304854870803468, −9.447327064971025872946981707894, −9.079602582034419851604444457941, −8.216138576673854447657607329486, −7.36089293516513615040819902033, −5.87746557115723513707045546827, −5.48775811313321755851411290680, −4.35662561433885184614702047190, −3.58762789047600305318352307541, −2.09500727755893704178264259859, 1.00203755457675997246144015494, 1.88363648473955196995397431421, 3.42673402640407648360477884543, 4.21386049318643692393114239504, 5.63536521934718264250805684396, 6.35961245996289163806563964937, 7.58252841875190013810646320869, 8.183520386992957744428482140487, 9.026565925050278983327245493720, 10.30211132750679263478848902049

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