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Properties

Label 2-300-20.3-c1-0-16
Degree $2$
Conductor $300$
Sign $-0.881 + 0.471i$
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  + (0.581 − 1.28i)2-s + (−0.707 + 0.707i)3-s + (−1.32 − 1.50i)4-s + (0.500 + 1.32i)6-s + (−1.41 − 1.41i)7-s + (−2.70 + 0.832i)8-s − 1.00i·9-s − 5.29i·11-s + (1.99 + 0.125i)12-s + (−3.74 − 3.74i)13-s + (−2.64 + 1.00i)14-s + (−0.5 + 3.96i)16-s + (−1.28 − 0.581i)18-s + 5.29·19-s + 2.00·21-s + (−6.82 − 3.07i)22-s + ⋯
L(s)  = 1  + (0.411 − 0.911i)2-s + (−0.408 + 0.408i)3-s + (−0.661 − 0.750i)4-s + (0.204 + 0.540i)6-s + (−0.534 − 0.534i)7-s + (−0.955 + 0.294i)8-s − 0.333i·9-s − 1.59i·11-s + (0.576 + 0.0361i)12-s + (−1.03 − 1.03i)13-s + (−0.707 + 0.267i)14-s + (−0.125 + 0.992i)16-s + (−0.303 − 0.137i)18-s + 1.21·19-s + 0.436·21-s + (−1.45 − 0.656i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.881 + 0.471i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.881 + 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228043 - 0.910331i\)
\(L(\frac12)\) \(\approx\) \(0.228043 - 0.910331i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.581 + 1.28i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (1.41 + 1.41i)T + 7iT^{2} \)
11 \( 1 + 5.29iT - 11T^{2} \)
13 \( 1 + (3.74 + 3.74i)T + 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 - 8iT - 29T^{2} \)
31 \( 1 + 5.29iT - 31T^{2} \)
37 \( 1 + (-3.74 + 3.74i)T - 37iT^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + (-5.65 + 5.65i)T - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-7.48 - 7.48i)T + 53iT^{2} \)
59 \( 1 + 5.29T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + (-8.48 - 8.48i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (7.48 + 7.48i)T + 73iT^{2} \)
79 \( 1 + 5.29T + 79T^{2} \)
83 \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + (-7.48 + 7.48i)T - 97iT^{2} \)
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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

−11.32133714762949500202879265536, −10.49608470973322400812445688612, −9.827862752689409637193911280504, −8.853753657203960796457193039296, −7.45971766363412715861701030001, −5.90512490069885545510684548930, −5.27238388607455772792031778286, −3.81327161103452755568936856090, −2.96486803733508353378127532663, −0.63511398975853039823259255291, 2.46974784227322696232601674249, 4.29864171522189442654944847211, 5.19113952404350885894387436936, 6.39614891026667465664209358085, 7.10318784973835083574842739297, 7.962944173410294225539948901509, 9.390407723287935901078127303132, 9.872743707304082871620040065404, 11.77276670574379259634913751633, 12.19225626828744473095159948614

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