Properties

Label 2-300-20.7-c1-0-4
Degree $2$
Conductor $300$
Sign $-0.619 - 0.785i$
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  + (−0.760 + 1.19i)2-s + (0.707 + 0.707i)3-s + (−0.844 − 1.81i)4-s + (−1.38 + 0.305i)6-s + (−0.611 + 0.611i)7-s + (2.80 + 0.371i)8-s + 1.00i·9-s + 5.12i·11-s + (0.685 − 1.87i)12-s + (−1.76 + 1.76i)13-s + (−0.264 − 1.19i)14-s + (−2.57 + 3.06i)16-s + (3.76 + 3.76i)17-s + (−1.19 − 0.760i)18-s − 1.22·19-s + ⋯
L(s)  = 1  + (−0.537 + 0.843i)2-s + (0.408 + 0.408i)3-s + (−0.422 − 0.906i)4-s + (−0.563 + 0.124i)6-s + (−0.231 + 0.231i)7-s + (0.991 + 0.131i)8-s + 0.333i·9-s + 1.54i·11-s + (0.197 − 0.542i)12-s + (−0.488 + 0.488i)13-s + (−0.0706 − 0.319i)14-s + (−0.643 + 0.765i)16-s + (0.912 + 0.912i)17-s + (−0.281 − 0.179i)18-s − 0.280·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.619 - 0.785i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.619 - 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.428104 + 0.882621i\)
\(L(\frac12)\) \(\approx\) \(0.428104 + 0.882621i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.760 - 1.19i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (0.611 - 0.611i)T - 7iT^{2} \)
11 \( 1 - 5.12iT - 11T^{2} \)
13 \( 1 + (1.76 - 1.76i)T - 13iT^{2} \)
17 \( 1 + (-3.76 - 3.76i)T + 17iT^{2} \)
19 \( 1 + 1.22T + 19T^{2} \)
23 \( 1 + (-1.07 - 1.07i)T + 23iT^{2} \)
29 \( 1 + 0.864iT - 29T^{2} \)
31 \( 1 + 7.81iT - 31T^{2} \)
37 \( 1 + (-1.76 - 1.76i)T + 37iT^{2} \)
41 \( 1 - 5.52T + 41T^{2} \)
43 \( 1 + (6.20 + 6.20i)T + 43iT^{2} \)
47 \( 1 + (2.29 - 2.29i)T - 47iT^{2} \)
53 \( 1 + (-2.62 + 2.62i)T - 53iT^{2} \)
59 \( 1 + 0.528T + 59T^{2} \)
61 \( 1 - 4.98T + 61T^{2} \)
67 \( 1 + (-6.20 + 6.20i)T - 67iT^{2} \)
71 \( 1 + 8.10iT - 71T^{2} \)
73 \( 1 + (-2.25 + 2.25i)T - 73iT^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 + (-7.95 - 7.95i)T + 83iT^{2} \)
89 \( 1 + 7.25iT - 89T^{2} \)
97 \( 1 + (0.793 + 0.793i)T + 97iT^{2} \)
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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

−12.14135191535420276293127015732, −10.76832197966024277674380336746, −9.763027637050141811950000327904, −9.434675700272127093293903100243, −8.170651096965625536654770312411, −7.40702433371978188272674840900, −6.32921943845144641275536257640, −5.10127093459004556360972450506, −4.06999611731451464439701201278, −2.04101588244523753159519780424, 0.866095057040005685096438766995, 2.74274511803492942063864279541, 3.57342429284896941660855445346, 5.23913012227238981134461460127, 6.78633712839045520103306452019, 7.86353827576345753589101572037, 8.596054214879879776513146489281, 9.556449922510788482394134511102, 10.47870184216390199949840616156, 11.37847855872475787686559494062

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