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Properties

Label 2-88445-1.1-c1-0-1
Degree $2$
Conductor $88445$
Sign $1$
Analytic cond. $706.236$
Root an. cond. $26.5751$
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  + 2·2-s − 2·3-s + 2·4-s − 5-s − 4·6-s + 9-s − 2·10-s − 5·11-s − 4·12-s + 2·15-s − 4·16-s + 4·17-s + 2·18-s − 2·20-s − 10·22-s − 6·23-s + 25-s + 4·27-s + 9·29-s + 4·30-s + 3·31-s − 8·32-s + 10·33-s + 8·34-s + 2·36-s − 2·37-s − 2·41-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s − 0.447·5-s − 1.63·6-s + 1/3·9-s − 0.632·10-s − 1.50·11-s − 1.15·12-s + 0.516·15-s − 16-s + 0.970·17-s + 0.471·18-s − 0.447·20-s − 2.13·22-s − 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.67·29-s + 0.730·30-s + 0.538·31-s − 1.41·32-s + 1.74·33-s + 1.37·34-s + 1/3·36-s − 0.328·37-s − 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88445\)    =    \(5 \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(706.236\)
Root analytic conductor: \(26.5751\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 88445,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.075865588\)
\(L(\frac12)\) \(\approx\) \(1.075865588\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.77843435138565, −13.49218614081653, −12.70732473128402, −12.33451978912596, −12.10700491549059, −11.68664095412401, −10.95882256004344, −10.77412861740043, −9.929802150826417, −9.858143025877349, −8.627913129921799, −8.224890321708749, −7.780952679010906, −6.963494916940166, −6.502563516049668, −6.055479930695892, −5.416926910759717, −5.092586518765860, −4.786152284299939, −3.988583372874985, −3.496528555266825, −2.753516483297482, −2.375814993661362, −1.201128928925290, −0.2948141511218981, 0.2948141511218981, 1.201128928925290, 2.375814993661362, 2.753516483297482, 3.496528555266825, 3.988583372874985, 4.786152284299939, 5.092586518765860, 5.416926910759717, 6.055479930695892, 6.502563516049668, 6.963494916940166, 7.780952679010906, 8.224890321708749, 8.627913129921799, 9.858143025877349, 9.929802150826417, 10.77412861740043, 10.95882256004344, 11.68664095412401, 12.10700491549059, 12.33451978912596, 12.70732473128402, 13.49218614081653, 13.77843435138565

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