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Properties

Label 2-750-125.89-c1-0-18
Degree $2$
Conductor $750$
Sign $0.134 + 0.990i$
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  + (0.481 − 0.876i)2-s + (−0.368 + 0.929i)3-s + (−0.535 − 0.844i)4-s + (1.84 − 1.26i)5-s + (0.637 + 0.770i)6-s + (1.92 + 0.625i)7-s + (−0.998 + 0.0627i)8-s + (−0.728 − 0.684i)9-s + (−0.221 − 2.22i)10-s + (−5.45 − 2.99i)11-s + (0.982 − 0.187i)12-s + (4.51 − 4.81i)13-s + (1.47 − 1.38i)14-s + (0.499 + 2.17i)15-s + (−0.425 + 0.904i)16-s + (0.488 + 0.310i)17-s + ⋯
L(s)  = 1  + (0.340 − 0.619i)2-s + (−0.212 + 0.536i)3-s + (−0.267 − 0.422i)4-s + (0.824 − 0.566i)5-s + (0.260 + 0.314i)6-s + (0.727 + 0.236i)7-s + (−0.352 + 0.0221i)8-s + (−0.242 − 0.228i)9-s + (−0.0702 − 0.703i)10-s + (−1.64 − 0.903i)11-s + (0.283 − 0.0540i)12-s + (1.25 − 1.33i)13-s + (0.394 − 0.370i)14-s + (0.128 + 0.562i)15-s + (−0.106 + 0.226i)16-s + (0.118 + 0.0751i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.134 + 0.990i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.134 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41759 - 1.23810i\)
\(L(\frac12)\) \(\approx\) \(1.41759 - 1.23810i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.481 + 0.876i)T \)
3 \( 1 + (0.368 - 0.929i)T \)
5 \( 1 + (-1.84 + 1.26i)T \)
good7 \( 1 + (-1.92 - 0.625i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (5.45 + 2.99i)T + (5.89 + 9.28i)T^{2} \)
13 \( 1 + (-4.51 + 4.81i)T + (-0.816 - 12.9i)T^{2} \)
17 \( 1 + (-0.488 - 0.310i)T + (7.23 + 15.3i)T^{2} \)
19 \( 1 + (0.442 - 0.175i)T + (13.8 - 13.0i)T^{2} \)
23 \( 1 + (-0.714 - 5.65i)T + (-22.2 + 5.71i)T^{2} \)
29 \( 1 + (-4.65 + 1.19i)T + (25.4 - 13.9i)T^{2} \)
31 \( 1 + (-2.62 + 4.13i)T + (-13.1 - 28.0i)T^{2} \)
37 \( 1 + (1.76 + 0.830i)T + (23.5 + 28.5i)T^{2} \)
41 \( 1 + (-3.16 - 0.399i)T + (39.7 + 10.1i)T^{2} \)
43 \( 1 + (-4.44 + 6.11i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (11.0 + 0.696i)T + (46.6 + 5.89i)T^{2} \)
53 \( 1 + (-3.54 - 2.93i)T + (9.93 + 52.0i)T^{2} \)
59 \( 1 + (-1.51 - 7.92i)T + (-54.8 + 21.7i)T^{2} \)
61 \( 1 + (4.65 - 0.588i)T + (59.0 - 15.1i)T^{2} \)
67 \( 1 + (-0.933 + 3.63i)T + (-58.7 - 32.2i)T^{2} \)
71 \( 1 + (0.278 - 4.42i)T + (-70.4 - 8.89i)T^{2} \)
73 \( 1 + (-8.65 - 1.65i)T + (67.8 + 26.8i)T^{2} \)
79 \( 1 + (-14.6 - 5.79i)T + (57.5 + 54.0i)T^{2} \)
83 \( 1 + (6.56 + 16.5i)T + (-60.5 + 56.8i)T^{2} \)
89 \( 1 + (1.83 - 9.61i)T + (-82.7 - 32.7i)T^{2} \)
97 \( 1 + (-2.44 - 9.53i)T + (-85.0 + 46.7i)T^{2} \)
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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

−10.47301008611518769893627551613, −9.501859132208553535233260021967, −8.448158577585382448963274988196, −8.010252175764505738164003386417, −6.02920359157787776670483623614, −5.51555897256212062172873137267, −4.91047623552895036025460984764, −3.53245593499403521983665621466, −2.49432126265001641908182252779, −0.950200551156991097087851035054, 1.71712367544391577188338912957, 2.84382354750811769959800181675, 4.50438116474999625479618273294, 5.21134600609354087900013100814, 6.37176635164273959237222972095, 6.82369128933698753230483351800, 7.85684883918810233830482709100, 8.549999004733817525665936957979, 9.719303515790856436906146842722, 10.68276277174300491594826294966

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