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Properties

Label 2-300-15.8-c1-0-3
Degree $2$
Conductor $300$
Sign $0.945 - 0.326i$
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  + (1.22 + 1.22i)3-s + (3.67 − 3.67i)7-s + 2.99i·9-s + (−1.22 − 1.22i)13-s + 7i·19-s + 9·21-s + (−3.67 + 3.67i)27-s − 11·31-s + (4.89 − 4.89i)37-s − 2.99i·39-s + (−1.22 − 1.22i)43-s − 20i·49-s + (−8.57 + 8.57i)57-s − 61-s + (11.0 + 11.0i)63-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (1.38 − 1.38i)7-s + 0.999i·9-s + (−0.339 − 0.339i)13-s + 1.60i·19-s + 1.96·21-s + (−0.707 + 0.707i)27-s − 1.97·31-s + (0.805 − 0.805i)37-s − 0.480i·39-s + (−0.186 − 0.186i)43-s − 2.85i·49-s + (−1.13 + 1.13i)57-s − 0.128·61-s + (1.38 + 1.38i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.945 - 0.326i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.945 - 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73555 + 0.291225i\)
\(L(\frac12)\) \(\approx\) \(1.73555 + 0.291225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (-3.67 + 3.67i)T - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (1.22 + 1.22i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 - 7iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 11T + 31T^{2} \)
37 \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (1.22 + 1.22i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + (8.57 - 8.57i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (9.79 + 9.79i)T + 73iT^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-3.67 + 3.67i)T - 97iT^{2} \)
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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

−11.52030785592762076557666464599, −10.64422132286676405041391494598, −10.13113842523790959479490727842, −8.915701980423007964693690494091, −7.84148658730117904538530503067, −7.43842888431618224300548340974, −5.52482853605190452122363885586, −4.44020035648509998984483148507, −3.61653648692917148728047209142, −1.77973688557852886147883827335, 1.78628838782571669735172043363, 2.80015991950373825266521744282, 4.60886312772752862324915921247, 5.71177491798444827396276226959, 7.00924955764891364215982017054, 7.953120423714196314595464752976, 8.822555355197016147019460866317, 9.377872132847531494493509109494, 11.09738498905289524772231121769, 11.72877412032451963904592078077

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