Properties

Label 2-5445-1.1-c1-0-68
Degree $2$
Conductor $5445$
Sign $1$
Analytic cond. $43.4785$
Root an. cond. $6.59382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·2-s − 1.82·4-s + 5-s + 2·7-s + 1.58·8-s − 0.414·10-s + 6.82·13-s − 0.828·14-s + 3·16-s + 1.17·17-s − 1.82·20-s − 2.82·23-s + 25-s − 2.82·26-s − 3.65·28-s + 7.65·29-s − 4.41·32-s − 0.485·34-s + 2·35-s + 3.65·37-s + 1.58·40-s + 6·41-s + 6·43-s + 1.17·46-s + 2.82·47-s − 3·49-s − 0.414·50-s + ⋯
L(s)  = 1  − 0.292·2-s − 0.914·4-s + 0.447·5-s + 0.755·7-s + 0.560·8-s − 0.130·10-s + 1.89·13-s − 0.221·14-s + 0.750·16-s + 0.284·17-s − 0.408·20-s − 0.589·23-s + 0.200·25-s − 0.554·26-s − 0.691·28-s + 1.42·29-s − 0.780·32-s − 0.0832·34-s + 0.338·35-s + 0.601·37-s + 0.250·40-s + 0.937·41-s + 0.914·43-s + 0.172·46-s + 0.412·47-s − 0.428·49-s − 0.0585·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5445\)    =    \(3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(43.4785\)
Root analytic conductor: \(6.59382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5445} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5445,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.965430476\)
\(L(\frac12)\) \(\approx\) \(1.965430476\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + 0.414T + 2T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
13 \( 1 - 6.82T + 13T^{2} \)
17 \( 1 - 1.17T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 - 7.65T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 3.65T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 0.343T + 53T^{2} \)
59 \( 1 - 9.65T + 59T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 + 4.48T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 6.82T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 9.31T + 89T^{2} \)
97 \( 1 + 7.65T + 97T^{2} \)
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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

−8.184634255285142865140303647380, −7.82935500034133719715214491057, −6.65921136390532920334326821949, −5.92011876838888545158631286331, −5.33720244303735127442092987412, −4.38992513739548570022790784955, −3.91424476464776678903641097137, −2.81908782308141833850290925488, −1.54984122971752724080285589921, −0.894211443976235578482143057558, 0.894211443976235578482143057558, 1.54984122971752724080285589921, 2.81908782308141833850290925488, 3.91424476464776678903641097137, 4.38992513739548570022790784955, 5.33720244303735127442092987412, 5.92011876838888545158631286331, 6.65921136390532920334326821949, 7.82935500034133719715214491057, 8.184634255285142865140303647380

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