Properties

Label 2-5415-1.1-c1-0-172
Degree $2$
Conductor $5415$
Sign $-1$
Analytic cond. $43.2389$
Root an. cond. $6.57563$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.414·2-s + 3-s − 1.82·4-s − 5-s + 0.414·6-s + 0.585·7-s − 1.58·8-s + 9-s − 0.414·10-s + 1.41·11-s − 1.82·12-s − 5.41·13-s + 0.242·14-s − 15-s + 3·16-s − 1.17·17-s + 0.414·18-s + 1.82·20-s + 0.585·21-s + 0.585·22-s + 7.65·23-s − 1.58·24-s + 25-s − 2.24·26-s + 27-s − 1.07·28-s + 9.07·29-s + ⋯
L(s)  = 1  + 0.292·2-s + 0.577·3-s − 0.914·4-s − 0.447·5-s + 0.169·6-s + 0.221·7-s − 0.560·8-s + 0.333·9-s − 0.130·10-s + 0.426·11-s − 0.527·12-s − 1.50·13-s + 0.0648·14-s − 0.258·15-s + 0.750·16-s − 0.284·17-s + 0.0976·18-s + 0.408·20-s + 0.127·21-s + 0.124·22-s + 1.59·23-s − 0.323·24-s + 0.200·25-s − 0.439·26-s + 0.192·27-s − 0.202·28-s + 1.68·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5415\)    =    \(3 \cdot 5 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(43.2389\)
Root analytic conductor: \(6.57563\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5415,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
19 \( 1 \)
good2 \( 1 - 0.414T + 2T^{2} \)
7 \( 1 - 0.585T + 7T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 + 5.41T + 13T^{2} \)
17 \( 1 + 1.17T + 17T^{2} \)
23 \( 1 - 7.65T + 23T^{2} \)
29 \( 1 - 9.07T + 29T^{2} \)
31 \( 1 + 6.48T + 31T^{2} \)
37 \( 1 + 11.0T + 37T^{2} \)
41 \( 1 - 7.41T + 41T^{2} \)
43 \( 1 - 0.585T + 43T^{2} \)
47 \( 1 - 0.343T + 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 - 5.65T + 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 4.48T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 - 4.24T + 97T^{2} \)
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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

−7.84030647463489509528378969560, −7.20623957952738097328225642550, −6.51042640688552916328272555743, −5.28523764752888411192216032583, −4.83062512784019982218284723993, −4.18560815109805599792706238699, −3.29954381672394456154043438029, −2.63466562260784894646833291263, −1.31406608014786935305373840393, 0, 1.31406608014786935305373840393, 2.63466562260784894646833291263, 3.29954381672394456154043438029, 4.18560815109805599792706238699, 4.83062512784019982218284723993, 5.28523764752888411192216032583, 6.51042640688552916328272555743, 7.20623957952738097328225642550, 7.84030647463489509528378969560

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