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Properties

Label 2-300-20.7-c1-0-8
Degree $2$
Conductor $300$
Sign $0.945 + 0.326i$
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  + (−1.35 − 0.394i)2-s + (0.707 + 0.707i)3-s + (1.68 + 1.07i)4-s + (−0.681 − 1.23i)6-s + (2.47 − 2.47i)7-s + (−1.87 − 2.11i)8-s + 1.00i·9-s − 3.02i·11-s + (0.437 + 1.95i)12-s + (−0.363 + 0.363i)13-s + (−4.34 + 2.38i)14-s + (1.70 + 3.61i)16-s + (2.36 + 2.36i)17-s + (0.394 − 1.35i)18-s + 4.95·19-s + ⋯
L(s)  = 1  + (−0.960 − 0.278i)2-s + (0.408 + 0.408i)3-s + (0.844 + 0.535i)4-s + (−0.278 − 0.505i)6-s + (0.936 − 0.936i)7-s + (−0.661 − 0.749i)8-s + 0.333i·9-s − 0.913i·11-s + (0.126 + 0.563i)12-s + (−0.100 + 0.100i)13-s + (−1.16 + 0.638i)14-s + (0.426 + 0.904i)16-s + (0.573 + 0.573i)17-s + (0.0929 − 0.320i)18-s + 1.13·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.945 + 0.326i$
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: \((2,\ 300,\ (\ :1/2),\ 0.945 + 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06876 - 0.179655i\)
\(L(\frac12)\) \(\approx\) \(1.06876 - 0.179655i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.394i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (-2.47 + 2.47i)T - 7iT^{2} \)
11 \( 1 + 3.02iT - 11T^{2} \)
13 \( 1 + (0.363 - 0.363i)T - 13iT^{2} \)
17 \( 1 + (-2.36 - 2.36i)T + 17iT^{2} \)
19 \( 1 - 4.95T + 19T^{2} \)
23 \( 1 + (0.900 + 0.900i)T + 23iT^{2} \)
29 \( 1 - 3.50iT - 29T^{2} \)
31 \( 1 + 3.85iT - 31T^{2} \)
37 \( 1 + (-0.363 - 0.363i)T + 37iT^{2} \)
41 \( 1 - 2.72T + 41T^{2} \)
43 \( 1 + (-3.92 - 3.92i)T + 43iT^{2} \)
47 \( 1 + (-5.85 + 5.85i)T - 47iT^{2} \)
53 \( 1 + (3.14 - 3.14i)T - 53iT^{2} \)
59 \( 1 + 8.68T + 59T^{2} \)
61 \( 1 + 15.2T + 61T^{2} \)
67 \( 1 + (3.92 - 3.92i)T - 67iT^{2} \)
71 \( 1 - 4.25iT - 71T^{2} \)
73 \( 1 + (9.28 - 9.28i)T - 73iT^{2} \)
79 \( 1 + 0.399T + 79T^{2} \)
83 \( 1 + (0.199 + 0.199i)T + 83iT^{2} \)
89 \( 1 - 4.28iT - 89T^{2} \)
97 \( 1 + (6.73 + 6.73i)T + 97iT^{2} \)
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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

−11.28439755864006631198438164170, −10.72671417065690201356871264959, −9.850507093678165727298686607573, −8.860403950505518784931935240847, −7.957184678843115801591620153132, −7.33488789669137896933322993869, −5.80989759040633638693640223415, −4.21878107137396146040423626644, −3.05331725450058106902687451682, −1.29473534620360346379554490425, 1.57815415673670337045910947944, 2.78363854866929455152207567159, 4.96764829586685069669063835069, 6.01203746415230980677721409915, 7.41355266962109644827617696790, 7.83044706594078733095501840087, 8.993322032807980673516301160957, 9.587890396574491917670995545487, 10.75253580751491894642576473411, 11.91066158673029333497514994119

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