Properties

Label 2-300-15.8-c1-0-0
Degree $2$
Conductor $300$
Sign $-0.130 - 0.991i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.130 - 0.991i$
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: \((2,\ 300,\ (\ :1/2),\ -0.130 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.368743 + 0.420575i\)
\(L(\frac12)\) \(\approx\) \(0.368743 + 0.420575i\)
\(L(\frac{3}{2})\) 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

−12.27220848654636882662166615928, −11.25691716045323270066672446965, −10.13204404431441702005750004303, −9.188105604863573297427055625240, −8.166704774242188692017644598861, −6.88498259536766895505327696159, −6.06431871183911066777028233098, −5.38283383068591988890539478739, −3.49490087929753217566214985753, −2.00105029732420890941231738876, 0.43487692495510207052947178929, 3.29679380227759339369699130625, 4.15313930557551341846262812536, 5.45229441999026992233723255918, 6.62190355994087854172886235568, 7.27063806508378292236530837756, 9.014468345573923964701132788590, 9.724336581674903305675141520519, 10.66620269302834580076559266176, 11.11860238171895224302109532218

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