Properties

Label 2-300-60.59-c1-0-3
Degree $2$
Conductor $300$
Sign $0.137 - 0.990i$
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  + (−1.28 − 0.599i)2-s + (−1.66 + 0.468i)3-s + (1.28 + 1.53i)4-s + (2.41 + 0.400i)6-s + 0.936·7-s + (−0.719 − 2.73i)8-s + (2.56 − 1.56i)9-s − 4.27·11-s + (−2.85 − 1.96i)12-s + 3.12i·13-s + (−1.19 − 0.561i)14-s + (−0.719 + 3.93i)16-s + 2·17-s + (−4.21 + 0.463i)18-s + 4.27i·19-s + ⋯
L(s)  = 1  + (−0.905 − 0.424i)2-s + (−0.962 + 0.270i)3-s + (0.640 + 0.768i)4-s + (0.986 + 0.163i)6-s + 0.353·7-s + (−0.254 − 0.967i)8-s + (0.853 − 0.520i)9-s − 1.28·11-s + (−0.824 − 0.566i)12-s + 0.866i·13-s + (−0.320 − 0.150i)14-s + (−0.179 + 0.983i)16-s + 0.485·17-s + (−0.994 + 0.109i)18-s + 0.979i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.137 - 0.990i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.137 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.350358 + 0.304934i\)
\(L(\frac12)\) \(\approx\) \(0.350358 + 0.304934i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 + 0.599i)T \)
3 \( 1 + (1.66 - 0.468i)T \)
5 \( 1 \)
good7 \( 1 - 0.936T + 7T^{2} \)
11 \( 1 + 4.27T + 11T^{2} \)
13 \( 1 - 3.12iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 4.27iT - 19T^{2} \)
23 \( 1 - 7.60iT - 23T^{2} \)
29 \( 1 - 5.12iT - 29T^{2} \)
31 \( 1 + 2.39iT - 31T^{2} \)
37 \( 1 - 3.12iT - 37T^{2} \)
41 \( 1 - 7.12iT - 41T^{2} \)
43 \( 1 + 1.46T + 43T^{2} \)
47 \( 1 - 0.936iT - 47T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 - 7.19T + 59T^{2} \)
61 \( 1 + 5.12T + 61T^{2} \)
67 \( 1 - 5.20T + 67T^{2} \)
71 \( 1 - 6.67T + 71T^{2} \)
73 \( 1 + 8.24iT - 73T^{2} \)
79 \( 1 + 9.06iT - 79T^{2} \)
83 \( 1 + 4.68iT - 83T^{2} \)
89 \( 1 - 6.24iT - 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−11.65517806255378354662483284110, −11.01816786250833728605920624200, −10.11794302918467545768527696107, −9.462632780516056093938878868859, −8.109457090395765433176817706059, −7.31518090221273726043894756195, −6.10048068934853312541560492808, −4.92340706773048346020437575434, −3.46411383995866646948469270351, −1.60225717654946378237550656546, 0.52152372064319832357001784615, 2.42290555944526883482553778518, 4.88818243299042257718981437875, 5.63961168951944359359124166375, 6.74522732493433913037175610099, 7.70030497642541242659908558288, 8.428736338923023931321647537318, 9.876751141264314757764451082069, 10.59570037846481569987800530637, 11.18377772419039195431854456539

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