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Properties

Label 2-750-25.14-c1-0-0
Degree $2$
Conductor $750$
Sign $-0.909 - 0.414i$
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 − 2.61i·7-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (−2.92 + 2.12i)11-s + (0.587 − 0.809i)12-s + (−3.80 + 5.23i)13-s + (2.11 − 1.53i)14-s + (−0.809 − 0.587i)16-s + (1.17 − 0.381i)17-s + 0.999i·18-s + (1.76 + 5.42i)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 − 0.989i·7-s + (−0.336 + 0.109i)8-s + (0.269 + 0.195i)9-s + (−0.882 + 0.641i)11-s + (0.169 − 0.233i)12-s + (−1.05 + 1.45i)13-s + (0.566 − 0.411i)14-s + (−0.202 − 0.146i)16-s + (0.285 − 0.0926i)17-s + 0.235i·18-s + (0.404 + 1.24i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.909 - 0.414i$
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: \((2,\ 750,\ (\ :1/2),\ -0.909 - 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.161901 + 0.745125i\)
\(L(\frac12)\) \(\approx\) \(0.161901 + 0.745125i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.61iT - 7T^{2} \)
11 \( 1 + (2.92 - 2.12i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.80 - 5.23i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.17 + 0.381i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.76 - 5.42i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.62 + 3.61i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.61 - 8.05i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.04 - 6.29i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.70 - 6.47i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.61 + 3.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.70iT - 43T^{2} \)
47 \( 1 + (1.62 + 0.527i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.98 + 0.645i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.92 + 2.12i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.23 - 1.62i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.45 - 0.472i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.70 + 5.25i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.07 + 2.85i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.73 + 5.34i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.04 - 0.663i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-2.85 + 2.07i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.21 - 1.04i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.46403569926843220185213147593, −10.16642246035637246115541656732, −8.929392254237502614082913648533, −7.74885808741264912284219728511, −7.15608376109466961799505245259, −6.51498741825244792026374254960, −5.20215820725134689293739627521, −4.65074000085071706121911803233, −3.52082426151307261650660749681, −1.83892639593018116143396785418, 0.34492879837652489671466619030, 2.37992339993748554405888608885, 3.18936421066405222038697161410, 4.65856512351778644318965959283, 5.53928684979010413715426739081, 5.90400325303420618695352332110, 7.44661109068162845309679368291, 8.266426208029563074354398854512, 9.511322658887647991519403448939, 10.00845200733101843811227135682

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