Properties

Degree $2$
Conductor $2015$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.709·2-s + 1.77·3-s − 0.497·4-s + 5-s − 1.25·6-s + 0.241·7-s + 1.06·8-s + 2.13·9-s − 0.709·10-s − 1.94·11-s − 0.880·12-s + 13-s − 0.170·14-s + 1.77·15-s − 0.255·16-s − 1.49·17-s − 1.51·18-s − 0.497·20-s + 0.426·21-s + 1.37·22-s + 1.13·23-s + 1.88·24-s + 25-s − 0.709·26-s + 2.01·27-s − 0.119·28-s − 1.25·30-s + ⋯
L(s)  = 1  − 0.709·2-s + 1.77·3-s − 0.497·4-s + 5-s − 1.25·6-s + 0.241·7-s + 1.06·8-s + 2.13·9-s − 0.709·10-s − 1.94·11-s − 0.880·12-s + 13-s − 0.170·14-s + 1.77·15-s − 0.255·16-s − 1.49·17-s − 1.51·18-s − 0.497·20-s + 0.426·21-s + 1.37·22-s + 1.13·23-s + 1.88·24-s + 25-s − 0.709·26-s + 2.01·27-s − 0.119·28-s − 1.25·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{2015} (2014, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2015,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.504290648\)
\(L(\frac12)\) \(\approx\) \(1.504290648\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
13 \( 1 - T \)
31 \( 1 - T \)
good2 \( 1 + 0.709T + T^{2} \)
3 \( 1 - 1.77T + T^{2} \)
7 \( 1 - 0.241T + T^{2} \)
11 \( 1 + 1.94T + T^{2} \)
17 \( 1 + 1.49T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.13T + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 0.709T + T^{2} \)
47 \( 1 - 1.13T + T^{2} \)
53 \( 1 + 1.94T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.94T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 0.241T + T^{2} \)
97 \( 1 - 1.77T + T^{2} \)
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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

−9.044389748343957138003795940184, −8.740512232549285952948624930630, −8.065475235388433584300601395890, −7.45181641134705319280880790941, −6.41117599428878867782463155304, −5.07712297812359709301636419625, −4.47311772906868840213179001396, −3.16577836995136424423978186844, −2.41675961549851593372635937691, −1.48863415467420407122357934209, 1.48863415467420407122357934209, 2.41675961549851593372635937691, 3.16577836995136424423978186844, 4.47311772906868840213179001396, 5.07712297812359709301636419625, 6.41117599428878867782463155304, 7.45181641134705319280880790941, 8.065475235388433584300601395890, 8.740512232549285952948624930630, 9.044389748343957138003795940184

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