Properties

Label 2-300-5.4-c1-0-1
Degree $2$
Conductor $300$
Sign $0.447 + 0.894i$
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  i·3-s i·7-s − 9-s + 6·11-s − 5i·13-s − 6i·17-s − 5·19-s − 21-s + 6i·23-s + i·27-s + 6·29-s − 31-s − 6i·33-s + 2i·37-s − 5·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 1.80·11-s − 1.38i·13-s − 1.45i·17-s − 1.14·19-s − 0.218·21-s + 1.25i·23-s + 0.192i·27-s + 1.11·29-s − 0.179·31-s − 1.04i·33-s + 0.328i·37-s − 0.800·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12912 - 0.697839i\)
\(L(\frac12)\) \(\approx\) \(1.12912 - 0.697839i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 + iT - 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 - 11iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 7iT - 97T^{2} \)
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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

−11.71306478234434015482116738004, −10.75249607353648839694489368388, −9.623017114521372743848486422670, −8.724756441585117969723478982766, −7.60226271306038584194672676009, −6.76323999610163864840394137867, −5.74533468177333226841672193430, −4.31922017549397217704922497208, −2.96711029637426858672282506267, −1.12681038010934193104071281744, 1.94481564146603068110691641447, 3.82815000832242875890834551587, 4.50818170756140184808639903899, 6.18398257820960945511517870165, 6.71286173103656458170818552512, 8.557247793109954747773763479036, 8.908658711272978093623584247637, 10.02826616092039816278803189986, 10.97390440400319704438721262985, 11.89775426919520103720214312700

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