Properties

Label 2-300-300.59-c1-0-16
Degree $2$
Conductor $300$
Sign $0.709 - 0.705i$
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  + (1.37 + 0.314i)2-s + (−1.65 − 0.514i)3-s + (1.80 + 0.866i)4-s + (−1.94 + 1.10i)5-s + (−2.11 − 1.22i)6-s + 2.63·7-s + (2.21 + 1.76i)8-s + (2.47 + 1.70i)9-s + (−3.02 + 0.906i)10-s + (0.908 + 0.660i)11-s + (−2.53 − 2.36i)12-s + (4.15 + 5.72i)13-s + (3.62 + 0.826i)14-s + (3.78 − 0.820i)15-s + (2.49 + 3.12i)16-s + (1.53 − 4.72i)17-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)2-s + (−0.954 − 0.296i)3-s + (0.901 + 0.433i)4-s + (−0.870 + 0.492i)5-s + (−0.865 − 0.501i)6-s + 0.994·7-s + (0.782 + 0.622i)8-s + (0.823 + 0.566i)9-s + (−0.958 + 0.286i)10-s + (0.273 + 0.199i)11-s + (−0.731 − 0.681i)12-s + (1.15 + 1.58i)13-s + (0.969 + 0.220i)14-s + (0.977 − 0.211i)15-s + (0.624 + 0.781i)16-s + (0.372 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.709 - 0.705i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.709 - 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60934 + 0.664044i\)
\(L(\frac12)\) \(\approx\) \(1.60934 + 0.664044i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.314i)T \)
3 \( 1 + (1.65 + 0.514i)T \)
5 \( 1 + (1.94 - 1.10i)T \)
good7 \( 1 - 2.63T + 7T^{2} \)
11 \( 1 + (-0.908 - 0.660i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-4.15 - 5.72i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.53 + 4.72i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (6.71 + 2.18i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.81 - 2.49i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.240 + 0.0782i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.71 + 1.53i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.57 + 3.53i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.13 + 2.93i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.40T + 43T^{2} \)
47 \( 1 + (-9.30 + 3.02i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.355 + 1.09i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.37 - 0.998i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.31 + 2.41i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.555 - 1.70i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.17 + 6.69i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.35 - 1.86i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (5.36 - 1.74i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (8.73 + 2.83i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-7.11 + 9.79i)T + (-27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.59 + 0.517i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.71390123297994498030219486514, −11.30380723499221954076891587236, −10.67495794932176963539918763000, −8.767901071257523055794876188077, −7.51068384489168853007553595400, −6.91189436955797039318968120378, −5.91986798342540324357080479274, −4.62230259136098892947915808949, −3.97938628483523325571570105207, −1.93907046131189090133295853669, 1.28742395061797384792721366845, 3.67028093903403324441096482401, 4.39793551052973499651215426116, 5.51338379286212757746321918431, 6.24755422024512654623779412653, 7.72374672623426273349059929672, 8.560877737765904058989046816561, 10.59322755655785814610715901619, 10.69312003088043827454566052386, 11.68653756027582344440999214444

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