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Properties

Label 2-300-75.17-c1-0-8
Degree $2$
Conductor $300$
Sign $-0.194 + 0.980i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
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  + (−0.351 − 1.69i)3-s + (2.22 − 0.184i)5-s + (−2.84 − 2.84i)7-s + (−2.75 + 1.19i)9-s + (4.71 − 1.53i)11-s + (−0.188 − 0.369i)13-s + (−1.09 − 3.71i)15-s + (−7.50 + 1.18i)17-s + (2.68 − 3.69i)19-s + (−3.81 + 5.81i)21-s + (0.153 − 0.301i)23-s + (4.93 − 0.821i)25-s + (2.98 + 4.25i)27-s + (0.836 − 0.607i)29-s + (−2.24 − 1.63i)31-s + ⋯
L(s)  = 1  + (−0.202 − 0.979i)3-s + (0.996 − 0.0824i)5-s + (−1.07 − 1.07i)7-s + (−0.917 + 0.397i)9-s + (1.42 − 0.461i)11-s + (−0.0522 − 0.102i)13-s + (−0.282 − 0.959i)15-s + (−1.82 + 0.288i)17-s + (0.616 − 0.848i)19-s + (−0.833 + 1.26i)21-s + (0.0320 − 0.0628i)23-s + (0.986 − 0.164i)25-s + (0.575 + 0.818i)27-s + (0.155 − 0.112i)29-s + (−0.403 − 0.292i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.194 + 0.980i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.194 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.779850 - 0.950123i\)
\(L(\frac12)\) \(\approx\) \(0.779850 - 0.950123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.351 + 1.69i)T \)
5 \( 1 + (-2.22 + 0.184i)T \)
good7 \( 1 + (2.84 + 2.84i)T + 7iT^{2} \)
11 \( 1 + (-4.71 + 1.53i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.188 + 0.369i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (7.50 - 1.18i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-2.68 + 3.69i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.153 + 0.301i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (-0.836 + 0.607i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.24 + 1.63i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-4.31 + 2.20i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-5.67 - 1.84i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-2.46 + 2.46i)T - 43iT^{2} \)
47 \( 1 + (1.21 - 7.64i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-8.21 - 1.30i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (3.09 - 9.53i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.00 - 9.24i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-1.15 - 7.29i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (1.25 + 1.73i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-12.7 - 6.49i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (4.46 + 6.14i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.02 + 6.47i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-4.13 - 12.7i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.19 - 0.348i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.43959655921057668980582700515, −10.67980736934001713235796440544, −9.439014941559842656955709379388, −8.830739229922597592329319463229, −7.21414342454189292617610044835, −6.59989095829010026350705082242, −5.89549412605860187882827483301, −4.19825589814840824301880164268, −2.61972046978436080712827878572, −0.976123706203028559119353625361, 2.32517025008377050531681469720, 3.67930430564347491587031767549, 5.01450078023512773290743282193, 6.12778114128889626509448439596, 6.66218616123791976780099518451, 8.776092021889220612398779262384, 9.369192531956295453034655530786, 9.807347304831909025045914841873, 11.01807855117824876588893954605, 11.94412612459506217541653159495

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