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Properties

Label 2-1755-195.194-c0-0-3
Degree $2$
Conductor $1755$
Sign $1$
Analytic cond. $0.875859$
Root an. cond. $0.935873$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 5-s − 7-s + 11-s + 13-s + 16-s + 17-s − 20-s − 2·23-s + 25-s − 28-s + 35-s + 2·37-s + 41-s + 44-s + 52-s + 53-s − 55-s + 59-s − 61-s + 64-s − 65-s − 67-s + 68-s + 71-s − 73-s − 77-s + ⋯
L(s)  = 1  + 4-s − 5-s − 7-s + 11-s + 13-s + 16-s + 17-s − 20-s − 2·23-s + 25-s − 28-s + 35-s + 2·37-s + 41-s + 44-s + 52-s + 53-s − 55-s + 59-s − 61-s + 64-s − 65-s − 67-s + 68-s + 71-s − 73-s − 77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.875859\)
Root analytic conductor: \(0.935873\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1755} (1754, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1755,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.215500066\)
\(L(\frac12)\) \(\approx\) \(1.215500066\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
13 \( 1 - T \)
good2 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T + T^{2} \)
11 \( 1 - T + T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 + T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )^{2} \)
41 \( 1 - T + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 - T + T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 + T )^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.623617684470645179710204695302, −8.561863973312229192605720514371, −7.82415945354555359994615888827, −7.14454208934427930784450773902, −6.19989567448538585623104218633, −5.90385383270571567536138601398, −4.14999859726320563186177311193, −3.63966009809376524029633266690, −2.71303839201666020002404187199, −1.21682364129694631928029988784, 1.21682364129694631928029988784, 2.71303839201666020002404187199, 3.63966009809376524029633266690, 4.14999859726320563186177311193, 5.90385383270571567536138601398, 6.19989567448538585623104218633, 7.14454208934427930784450773902, 7.82415945354555359994615888827, 8.561863973312229192605720514371, 9.623617684470645179710204695302

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