Properties

Label 2-7650-1.1-c1-0-89
Degree $2$
Conductor $7650$
Sign $1$
Analytic cond. $61.0855$
Root an. cond. $7.81572$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 4·7-s + 8-s + 2·11-s + 6·13-s + 4·14-s + 16-s − 17-s + 4·19-s + 2·22-s + 5·23-s + 6·26-s + 4·28-s + 10·31-s + 32-s − 34-s − 9·37-s + 4·38-s − 11·41-s − 10·43-s + 2·44-s + 5·46-s + 8·47-s + 9·49-s + 6·52-s − 11·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 0.603·11-s + 1.66·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.917·19-s + 0.426·22-s + 1.04·23-s + 1.17·26-s + 0.755·28-s + 1.79·31-s + 0.176·32-s − 0.171·34-s − 1.47·37-s + 0.648·38-s − 1.71·41-s − 1.52·43-s + 0.301·44-s + 0.737·46-s + 1.16·47-s + 9/7·49-s + 0.832·52-s − 1.51·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7650\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(61.0855\)
Root analytic conductor: \(7.81572\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.058224005\)
\(L(\frac12)\) \(\approx\) \(5.058224005\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
17 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 11 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−7.985537860823507319364059454113, −6.88933726630694787909884555467, −6.61812811922214293051626196442, −5.51941040798251481003769363914, −5.12330525859265529469069967000, −4.33977824292271141341339784349, −3.64247995417089730355005364511, −2.85998336048520689944133568239, −1.59470488278465744334536289089, −1.21420924893543569221885428019, 1.21420924893543569221885428019, 1.59470488278465744334536289089, 2.85998336048520689944133568239, 3.64247995417089730355005364511, 4.33977824292271141341339784349, 5.12330525859265529469069967000, 5.51941040798251481003769363914, 6.61812811922214293051626196442, 6.88933726630694787909884555467, 7.985537860823507319364059454113

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