Properties

Label 2-1215-45.29-c0-0-5
Degree $2$
Conductor $1215$
Sign $-0.939 - 0.342i$
Analytic cond. $0.606363$
Root an. cond. $0.778693$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.939 − 1.62i)2-s + (−1.26 − 2.19i)4-s + (−0.5 − 0.866i)5-s − 2.87·8-s − 1.87·10-s + (−1.43 + 2.49i)16-s + 0.347·17-s + 0.347·19-s + (−1.26 + 2.19i)20-s + (−0.766 − 1.32i)23-s + (−0.499 + 0.866i)25-s + (0.939 + 1.62i)31-s + (1.26 + 2.19i)32-s + (0.326 − 0.565i)34-s + (0.326 − 0.565i)38-s + ⋯
L(s)  = 1  + (0.939 − 1.62i)2-s + (−1.26 − 2.19i)4-s + (−0.5 − 0.866i)5-s − 2.87·8-s − 1.87·10-s + (−1.43 + 2.49i)16-s + 0.347·17-s + 0.347·19-s + (−1.26 + 2.19i)20-s + (−0.766 − 1.32i)23-s + (−0.499 + 0.866i)25-s + (0.939 + 1.62i)31-s + (1.26 + 2.19i)32-s + (0.326 − 0.565i)34-s + (0.326 − 0.565i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1215\)    =    \(3^{5} \cdot 5\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(0.606363\)
Root analytic conductor: \(0.778693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1215} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1215,\ (\ :0),\ -0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.332133936\)
\(L(\frac12)\) \(\approx\) \(1.332133936\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - 0.347T + T^{2} \)
19 \( 1 - 0.347T + T^{2} \)
23 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - 1.53T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.867522852847059923464483018728, −8.845421706859847399567282464976, −8.231430168899691163926814612751, −6.76519808403005850457485730661, −5.57619555562649377205914239020, −4.91064720073111717441306637375, −4.12743097976949670739868866370, −3.31461914421157489832965251043, −2.15744692020729398935865247413, −0.915388144085387702778319729450, 2.77737061251395926004580582686, 3.76125756602634345653048913374, 4.41897988616347564626738899807, 5.64514839872994527615271212167, 6.11992116875315679826834272891, 7.13041826432147015770932999067, 7.62738967628092990669454838971, 8.248255734400945144971980599183, 9.331575633936871911714801604332, 10.24114248022359554978059938147

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