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Properties

Label 2-227850-1.1-c1-0-105
Degree $2$
Conductor $227850$
Sign $-1$
Analytic cond. $1819.39$
Root an. cond. $42.6543$
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 + 8-s + 9-s − 4·11-s − 12-s − 4·13-s + 16-s − 6·17-s + 18-s − 4·22-s − 24-s − 4·26-s − 27-s + 4·29-s − 31-s + 32-s + 4·33-s − 6·34-s + 36-s − 4·37-s + 4·39-s + 6·41-s + 8·43-s − 4·44-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.852·22-s − 0.204·24-s − 0.784·26-s − 0.192·27-s + 0.742·29-s − 0.179·31-s + 0.176·32-s + 0.696·33-s − 1.02·34-s + 1/6·36-s − 0.657·37-s + 0.640·39-s + 0.937·41-s + 1.21·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 227850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(227850\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 31\)
Sign: $-1$
Analytic conductor: \(1819.39\)
Root analytic conductor: \(42.6543\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 227850,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
31 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.11889268451424, −12.71303446434439, −12.32604255691775, −11.88077279847546, −11.27798089461650, −10.92121698451834, −10.57884131693311, −9.960729507183488, −9.657813052068305, −8.892915269685332, −8.466670973427074, −7.772909892056431, −7.326421441480684, −7.053796004694905, −6.231435797886091, −6.083471776557163, −5.325136365604348, −4.878267960554994, −4.582930916893987, −4.091738662478755, −3.197592279624541, −2.772235242089721, −2.193390830401888, −1.702347792048421, −0.6427696588379394, 0, 0.6427696588379394, 1.702347792048421, 2.193390830401888, 2.772235242089721, 3.197592279624541, 4.091738662478755, 4.582930916893987, 4.878267960554994, 5.325136365604348, 6.083471776557163, 6.231435797886091, 7.053796004694905, 7.326421441480684, 7.772909892056431, 8.466670973427074, 8.892915269685332, 9.657813052068305, 9.960729507183488, 10.57884131693311, 10.92121698451834, 11.27798089461650, 11.88077279847546, 12.32604255691775, 12.71303446434439, 13.11889268451424

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