Properties

Label 2-5415-1.1-c1-0-27
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·2-s − 3-s + 3.82·4-s − 5-s + 2.41·6-s − 1.41·7-s − 4.41·8-s + 9-s + 2.41·10-s − 2.24·11-s − 3.82·12-s + 3.41·13-s + 3.41·14-s + 15-s + 2.99·16-s + 1.17·17-s − 2.41·18-s − 3.82·20-s + 1.41·21-s + 5.41·22-s + 7.65·23-s + 4.41·24-s + 25-s − 8.24·26-s − 27-s − 5.41·28-s − 1.41·29-s + ⋯
L(s)  = 1  − 1.70·2-s − 0.577·3-s + 1.91·4-s − 0.447·5-s + 0.985·6-s − 0.534·7-s − 1.56·8-s + 0.333·9-s + 0.763·10-s − 0.676·11-s − 1.10·12-s + 0.946·13-s + 0.912·14-s + 0.258·15-s + 0.749·16-s + 0.284·17-s − 0.569·18-s − 0.856·20-s + 0.308·21-s + 1.15·22-s + 1.59·23-s + 0.901·24-s + 0.200·25-s − 1.61·26-s − 0.192·27-s − 1.02·28-s − 0.262·29-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: $\chi_{5415} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5415,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4988150845\)
\(L(\frac12)\) \(\approx\) \(0.4988150845\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.41T + 2T^{2} \)
7 \( 1 + 1.41T + 7T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
13 \( 1 - 3.41T + 13T^{2} \)
17 \( 1 - 1.17T + 17T^{2} \)
23 \( 1 - 7.65T + 23T^{2} \)
29 \( 1 + 1.41T + 29T^{2} \)
31 \( 1 - 3.17T + 31T^{2} \)
37 \( 1 - 3.41T + 37T^{2} \)
41 \( 1 - 0.242T + 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + 7.65T + 47T^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 - 7.31T + 61T^{2} \)
67 \( 1 - 9.65T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 7.65T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + 5.89T + 89T^{2} \)
97 \( 1 - 9.75T + 97T^{2} \)
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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

−7.961297822617072408070681644727, −7.84721208835245750560075965943, −6.79235367677849492814735616609, −6.44410648581459075230236521785, −5.50432090076571703867907081640, −4.57090435077098425679869837570, −3.41621185245044703199838105347, −2.61154525792176734515021869087, −1.35762270765441350899846327149, −0.55654924884430229056701803874, 0.55654924884430229056701803874, 1.35762270765441350899846327149, 2.61154525792176734515021869087, 3.41621185245044703199838105347, 4.57090435077098425679869837570, 5.50432090076571703867907081640, 6.44410648581459075230236521785, 6.79235367677849492814735616609, 7.84721208835245750560075965943, 7.961297822617072408070681644727

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