Properties

Label 2-300-15.8-c1-0-4
Degree $2$
Conductor $300$
Sign $-0.229 + 0.973i$
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 + (2.44 − 2.44i)7-s + 2.99i·9-s + (−4.89 − 4.89i)13-s − 8i·19-s − 5.99·21-s + (3.67 − 3.67i)27-s + 4·31-s + (−4.89 + 4.89i)37-s + 11.9i·39-s + (7.34 + 7.34i)43-s − 4.99i·49-s + (−9.79 + 9.79i)57-s + 14·61-s + (7.34 + 7.34i)63-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (0.925 − 0.925i)7-s + 0.999i·9-s + (−1.35 − 1.35i)13-s − 1.83i·19-s − 1.30·21-s + (0.707 − 0.707i)27-s + 0.718·31-s + (−0.805 + 0.805i)37-s + 1.92i·39-s + (1.12 + 1.12i)43-s − 0.714i·49-s + (−1.29 + 1.29i)57-s + 1.79·61-s + (0.925 + 0.925i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.229 + 0.973i$
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: \((2,\ 300,\ (\ :1/2),\ -0.229 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.600575 - 0.758860i\)
\(L(\frac12)\) \(\approx\) \(0.600575 - 0.758860i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.44 + 2.44i)T - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (4.89 + 4.89i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (4.89 - 4.89i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-7.34 - 7.34i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + (-2.44 + 2.44i)T - 67iT^{2} \)
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 \( 1 + (9.79 - 9.79i)T - 97iT^{2} \)
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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

−11.39805598580936664178268936471, −10.75761051569832320141710226072, −9.851379432190725628770411968653, −8.271115255958706346250263979573, −7.50798738926167795514168174967, −6.76857922228371503159190183850, −5.29340370255185237460372487794, −4.62939666683130432817846544659, −2.59330614551833375819427227728, −0.799160189443931801462285114043, 2.07065228059845667830178901591, 3.97975442981480372751068180553, 5.00657973306111747453348303330, 5.82290054571889066703307141897, 7.06683226675601598557856900859, 8.353456367036923359828498620284, 9.330686424724207785598702588895, 10.13139948808572754283610991094, 11.17994107342761616850174790370, 12.10021391994180247394508129222

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