Properties

Label 2-300-300.71-c1-0-36
Degree $2$
Conductor $300$
Sign $0.558 - 0.829i$
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  + (1.16 + 0.799i)2-s + (1.51 + 0.835i)3-s + (0.721 + 1.86i)4-s + (0.0912 − 2.23i)5-s + (1.10 + 2.18i)6-s − 1.22i·7-s + (−0.649 + 2.75i)8-s + (1.60 + 2.53i)9-s + (1.89 − 2.53i)10-s + (−1.19 + 3.68i)11-s + (−0.463 + 3.43i)12-s + (−1.75 − 5.41i)13-s + (0.982 − 1.43i)14-s + (2.00 − 3.31i)15-s + (−2.95 + 2.69i)16-s + (0.726 + 0.999i)17-s + ⋯
L(s)  = 1  + (0.824 + 0.565i)2-s + (0.875 + 0.482i)3-s + (0.360 + 0.932i)4-s + (0.0408 − 0.999i)5-s + (0.449 + 0.893i)6-s − 0.464i·7-s + (−0.229 + 0.973i)8-s + (0.534 + 0.845i)9-s + (0.598 − 0.801i)10-s + (−0.361 + 1.11i)11-s + (−0.133 + 0.990i)12-s + (−0.487 − 1.50i)13-s + (0.262 − 0.383i)14-s + (0.517 − 0.855i)15-s + (−0.739 + 0.672i)16-s + (0.176 + 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.558 - 0.829i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.558 - 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21454 + 1.17814i\)
\(L(\frac12)\) \(\approx\) \(2.21454 + 1.17814i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 - 0.799i)T \)
3 \( 1 + (-1.51 - 0.835i)T \)
5 \( 1 + (-0.0912 + 2.23i)T \)
good7 \( 1 + 1.22iT - 7T^{2} \)
11 \( 1 + (1.19 - 3.68i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.75 + 5.41i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.726 - 0.999i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.675 + 0.930i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.01 + 3.13i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.27 - 4.50i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (6.00 + 8.27i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.51 - 4.66i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.19 + 0.711i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.85iT - 43T^{2} \)
47 \( 1 + (-7.46 - 5.42i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.27 - 5.87i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.19 + 6.76i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.50 + 4.62i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-7.52 - 10.3i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-1.92 - 1.39i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.789 + 2.42i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.96 - 4.08i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.96 + 4.33i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (11.9 + 3.89i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-5.45 - 3.96i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.55975261707827075723283057574, −10.90027698368520267094932737459, −9.906555824443718800160134027213, −8.871611473877676290832292337642, −7.86275557540546211658526548312, −7.33497087975861833527510090008, −5.55361459735588817444496018488, −4.74040357326877520316476935424, −3.81202936258744877005848376467, −2.37261651421053695687132406798, 1.97674501625503597195362770324, 2.98260737076213530756056811179, 3.96255034721550506672930025481, 5.64460920646736081435575330589, 6.64843068716668872668785583100, 7.50946727589471249120815123888, 8.964662924846679996502369253355, 9.737794928935729493819154733158, 10.93935648258867191439991993689, 11.65174076245936830325514107600

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