Properties

Label 2-300-20.3-c1-0-6
Degree $2$
Conductor $300$
Sign $0.189 - 0.981i$
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  + (0.707 + 1.22i)2-s + (0.707 − 0.707i)3-s + (−0.999 + 1.73i)4-s + (1.36 + 0.366i)6-s + (1.74 + 1.74i)7-s − 2.82·8-s − 1.00i·9-s + 2i·11-s + (0.517 + 1.93i)12-s + (4.05 + 4.05i)13-s + (−0.901 + 3.36i)14-s + (−2.00 − 3.46i)16-s + (4.24 − 4.24i)17-s + (1.22 − 0.707i)18-s − 7.19·19-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s + (0.408 − 0.408i)3-s + (−0.499 + 0.866i)4-s + (0.557 + 0.149i)6-s + (0.658 + 0.658i)7-s − 0.999·8-s − 0.333i·9-s + 0.603i·11-s + (0.149 + 0.557i)12-s + (1.12 + 1.12i)13-s + (−0.241 + 0.899i)14-s + (−0.500 − 0.866i)16-s + (1.02 − 1.02i)17-s + (0.288 − 0.166i)18-s − 1.65·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.189 - 0.981i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43365 + 1.18289i\)
\(L(\frac12)\) \(\approx\) \(1.43365 + 1.18289i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (-1.74 - 1.74i)T + 7iT^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + (-4.05 - 4.05i)T + 13iT^{2} \)
17 \( 1 + (-4.24 + 4.24i)T - 17iT^{2} \)
19 \( 1 + 7.19T + 19T^{2} \)
23 \( 1 + (-0.378 + 0.378i)T - 23iT^{2} \)
29 \( 1 + 7.46iT - 29T^{2} \)
31 \( 1 - 0.267iT - 31T^{2} \)
37 \( 1 + (-2.07 + 2.07i)T - 37iT^{2} \)
41 \( 1 + 5.46T + 41T^{2} \)
43 \( 1 + (1.74 - 1.74i)T - 43iT^{2} \)
47 \( 1 + (9.14 + 9.14i)T + 47iT^{2} \)
53 \( 1 + (-1.03 - 1.03i)T + 53iT^{2} \)
59 \( 1 - 4.53T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 + (-8.81 - 8.81i)T + 67iT^{2} \)
71 \( 1 + 9.46iT - 71T^{2} \)
73 \( 1 + (2.82 + 2.82i)T + 73iT^{2} \)
79 \( 1 + 0.535T + 79T^{2} \)
83 \( 1 + (-0.656 + 0.656i)T - 83iT^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + (4.43 - 4.43i)T - 97iT^{2} \)
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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

−12.03500718939647003412574885814, −11.45431611468202200783809683763, −9.751592232672664418176110646960, −8.708051383734173708209760536110, −8.146402906971523203432126502685, −6.99851954964959308171849961511, −6.15801581513089929057673338337, −4.93870917422369993105520452417, −3.82709420516811561868415359340, −2.19322927879178443119481888956, 1.40934423812448630895860693736, 3.20509406619514771883719907166, 4.01907419273732841686654262399, 5.23688680675532335963240735048, 6.30093874075467404149450173703, 8.124266647010406546289255559276, 8.640710963359188980863588637928, 10.07776759583067755214457455421, 10.70279637453085339646291086548, 11.23416275624515828710721085407

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