Properties

Label 2-277350-1.1-c1-0-78
Degree $2$
Conductor $277350$
Sign $-1$
Analytic cond. $2214.65$
Root an. cond. $47.0600$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s − 3·11-s + 12-s + 4·13-s − 2·14-s + 16-s + 3·17-s − 18-s + 19-s + 2·21-s + 3·22-s + 6·23-s − 24-s − 4·26-s + 27-s + 2·28-s − 6·29-s − 4·31-s − 32-s − 3·33-s − 3·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.229·19-s + 0.436·21-s + 0.639·22-s + 1.25·23-s − 0.204·24-s − 0.784·26-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.522·33-s − 0.514·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(277350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 43^{2}\)
Sign: $-1$
Analytic conductor: \(2214.65\)
Root analytic conductor: \(47.0600\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{277350} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 277350,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 \)
43 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−12.94397521495234, −12.72607349191220, −11.80700826320253, −11.59430835406439, −11.03858056026731, −10.61579785645907, −10.32885931102558, −9.670365380719190, −9.131018009977011, −8.914844027401367, −8.252824410521601, −7.972968098977521, −7.509344572423374, −7.146222371417499, −6.491359980315252, −5.865746534808366, −5.397761271488125, −4.957774160227347, −4.289298013847934, −3.490888061710840, −3.338832421113254, −2.509529594664461, −2.073347448491814, −1.313124953239076, −0.9961596351293957, 0, 0.9961596351293957, 1.313124953239076, 2.073347448491814, 2.509529594664461, 3.338832421113254, 3.490888061710840, 4.289298013847934, 4.957774160227347, 5.397761271488125, 5.865746534808366, 6.491359980315252, 7.146222371417499, 7.509344572423374, 7.972968098977521, 8.252824410521601, 8.914844027401367, 9.131018009977011, 9.670365380719190, 10.32885931102558, 10.61579785645907, 11.03858056026731, 11.59430835406439, 11.80700826320253, 12.72607349191220, 12.94397521495234

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