Properties

Label 2-1775-71.70-c0-0-5
Degree $2$
Conductor $1775$
Sign $1$
Analytic cond. $0.885840$
Root an. cond. $0.941190$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.24·2-s + 1.80·3-s + 0.554·4-s − 2.24·6-s + 0.554·8-s + 2.24·9-s + 0.999·12-s − 1.24·16-s − 2.80·18-s + 1.24·19-s + 1.00·24-s + 2.24·27-s − 1.80·29-s + 0.999·32-s + 1.24·36-s − 1.24·37-s − 1.55·38-s + 0.445·43-s − 2.24·48-s + 49-s − 2.80·54-s + 2.24·57-s + 2.24·58-s + 71-s + 1.24·72-s + 0.445·73-s + 1.55·74-s + ⋯
L(s)  = 1  − 1.24·2-s + 1.80·3-s + 0.554·4-s − 2.24·6-s + 0.554·8-s + 2.24·9-s + 0.999·12-s − 1.24·16-s − 2.80·18-s + 1.24·19-s + 1.00·24-s + 2.24·27-s − 1.80·29-s + 0.999·32-s + 1.24·36-s − 1.24·37-s − 1.55·38-s + 0.445·43-s − 2.24·48-s + 49-s − 2.80·54-s + 2.24·57-s + 2.24·58-s + 71-s + 1.24·72-s + 0.445·73-s + 1.55·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1775\)    =    \(5^{2} \cdot 71\)
Sign: $1$
Analytic conductor: \(0.885840\)
Root analytic conductor: \(0.941190\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1775} (851, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1775,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.134288806\)
\(L(\frac12)\) \(\approx\) \(1.134288806\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
71 \( 1 - T \)
good2 \( 1 + 1.24T + T^{2} \)
3 \( 1 - 1.80T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.24T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.80T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.24T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 0.445T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
73 \( 1 - 0.445T + T^{2} \)
79 \( 1 + 0.445T + T^{2} \)
83 \( 1 + 1.24T + T^{2} \)
89 \( 1 + 1.80T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.330132796341729703608864420600, −8.788800985474931443322493569254, −8.092999675564426862480072422022, −7.44634167506164238076235736069, −6.98849590131557118033005282891, −5.40012912993915675875647282975, −4.18730425919183222403248140492, −3.37656538460639035528715631921, −2.30469465717266151622483926916, −1.39693348684821972064541686038, 1.39693348684821972064541686038, 2.30469465717266151622483926916, 3.37656538460639035528715631921, 4.18730425919183222403248140492, 5.40012912993915675875647282975, 6.98849590131557118033005282891, 7.44634167506164238076235736069, 8.092999675564426862480072422022, 8.788800985474931443322493569254, 9.330132796341729703608864420600

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