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Properties

Label 2-300-100.23-c1-0-0
Degree $2$
Conductor $300$
Sign $0.777 - 0.628i$
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  + (−0.970 − 1.02i)2-s + (−0.453 − 0.891i)3-s + (−0.115 + 1.99i)4-s + (−0.872 + 2.05i)5-s + (−0.475 + 1.33i)6-s + (−3.13 − 3.13i)7-s + (2.16 − 1.81i)8-s + (−0.587 + 0.809i)9-s + (2.96 − 1.10i)10-s + (3.28 + 4.52i)11-s + (1.83 − 0.803i)12-s + (0.499 + 3.15i)13-s + (−0.180 + 6.25i)14-s + (2.23 − 0.156i)15-s + (−3.97 − 0.460i)16-s + (2.24 + 1.14i)17-s + ⋯
L(s)  = 1  + (−0.686 − 0.727i)2-s + (−0.262 − 0.514i)3-s + (−0.0576 + 0.998i)4-s + (−0.390 + 0.920i)5-s + (−0.194 + 0.543i)6-s + (−1.18 − 1.18i)7-s + (0.765 − 0.643i)8-s + (−0.195 + 0.269i)9-s + (0.937 − 0.348i)10-s + (0.990 + 1.36i)11-s + (0.528 − 0.232i)12-s + (0.138 + 0.875i)13-s + (−0.0482 + 1.67i)14-s + (0.575 − 0.0405i)15-s + (−0.993 − 0.115i)16-s + (0.543 + 0.276i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.777 - 0.628i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.777 - 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.527170 + 0.186389i\)
\(L(\frac12)\) \(\approx\) \(0.527170 + 0.186389i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.970 + 1.02i)T \)
3 \( 1 + (0.453 + 0.891i)T \)
5 \( 1 + (0.872 - 2.05i)T \)
good7 \( 1 + (3.13 + 3.13i)T + 7iT^{2} \)
11 \( 1 + (-3.28 - 4.52i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.499 - 3.15i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-2.24 - 1.14i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.732 - 2.25i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.33 - 8.40i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-3.85 - 1.25i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.06 + 0.670i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.19 - 0.506i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (5.37 + 3.90i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-1.00 + 1.00i)T - 43iT^{2} \)
47 \( 1 + (-0.610 + 0.311i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (8.35 - 4.25i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (3.98 + 2.89i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.67 - 1.94i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.883 - 1.73i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (7.28 + 2.36i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.69 - 1.21i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (1.42 - 4.39i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-11.1 - 5.69i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-7.91 - 10.8i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.0175 - 0.0345i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.91900912906036668049619245600, −10.82836091584508514373054071695, −9.993530523687878858222361776970, −9.423379300227235115158449249074, −7.78494346453405199570452035484, −7.06471081931682406989077225783, −6.52454835369893671223685794761, −4.13260364697744434276756299800, −3.36237794089577169956284814737, −1.59908595395271377524514413991, 0.55927565535777128434724376780, 3.18839888310749951339273269513, 4.83144535200808179938660059451, 5.88393357780128849394210473391, 6.46480626244486117774771637970, 8.200461203939150320437379705809, 8.802062035493381040030410088659, 9.455061986073695903242238796436, 10.45024650259219496755233034606, 11.63008889442204423848060529188

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