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Properties

Label 2-300-300.239-c1-0-38
Degree $2$
Conductor $300$
Sign $0.913 - 0.406i$
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  + (−0.117 + 1.40i)2-s + (1.44 − 0.957i)3-s + (−1.97 − 0.331i)4-s + (2.16 − 0.555i)5-s + (1.18 + 2.14i)6-s − 0.146·7-s + (0.699 − 2.74i)8-s + (1.16 − 2.76i)9-s + (0.527 + 3.11i)10-s + (0.745 − 0.541i)11-s + (−3.16 + 1.41i)12-s + (0.637 − 0.876i)13-s + (0.0172 − 0.207i)14-s + (2.59 − 2.87i)15-s + (3.78 + 1.30i)16-s + (0.898 + 2.76i)17-s + ⋯
L(s)  = 1  + (−0.0831 + 0.996i)2-s + (0.833 − 0.552i)3-s + (−0.986 − 0.165i)4-s + (0.968 − 0.248i)5-s + (0.481 + 0.876i)6-s − 0.0555·7-s + (0.247 − 0.968i)8-s + (0.388 − 0.921i)9-s + (0.166 + 0.985i)10-s + (0.224 − 0.163i)11-s + (−0.913 + 0.407i)12-s + (0.176 − 0.243i)13-s + (0.00461 − 0.0553i)14-s + (0.669 − 0.742i)15-s + (0.945 + 0.326i)16-s + (0.218 + 0.670i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.913 - 0.406i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.913 - 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64858 + 0.350122i\)
\(L(\frac12)\) \(\approx\) \(1.64858 + 0.350122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.117 - 1.40i)T \)
3 \( 1 + (-1.44 + 0.957i)T \)
5 \( 1 + (-2.16 + 0.555i)T \)
good7 \( 1 + 0.146T + 7T^{2} \)
11 \( 1 + (-0.745 + 0.541i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.637 + 0.876i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.898 - 2.76i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.30 - 1.72i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.697 - 0.959i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-8.42 - 2.73i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.35 - 1.73i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.39 - 3.29i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.60 - 7.71i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.69T + 43T^{2} \)
47 \( 1 + (7.15 + 2.32i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.73 + 5.34i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.62 - 6.26i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.04 + 2.21i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.55 + 7.86i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.75 + 5.39i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (9.19 + 12.6i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-14.8 - 4.81i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (13.5 - 4.40i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-1.57 - 2.16i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.03 - 1.63i)T + (78.4 + 57.0i)T^{2} \)
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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

−12.27935835285459827337013452202, −10.42854120549079085876731921296, −9.665868649000638863049016242240, −8.610049283633959991413199701226, −8.213710095161394089235400341529, −6.75501411645601708842456615404, −6.23418139886818297066136360196, −4.91249516838330640864923792713, −3.42739657779202323796142413147, −1.56432196630702921524939153506, 1.93202139224104439125678722618, 2.94576239322833203990794153157, 4.22628604444424894286246548027, 5.29568270624724709231704917326, 6.86883189453239248973968553086, 8.373969653068497831075774124974, 9.067692462263980507666734221570, 9.915053534709213254361079986662, 10.50154356362540368847825087398, 11.47988351058770272773221465995

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