Properties

Degree 2
Conductor $ 5 \cdot 13 \cdot 31 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.13·2-s + 1.94·3-s + 0.290·4-s + 5-s + 2.20·6-s − 1.49·7-s − 0.805·8-s + 2.77·9-s + 1.13·10-s − 0.241·11-s + 0.564·12-s − 13-s − 1.70·14-s + 1.94·15-s − 1.20·16-s + 0.709·17-s + 3.14·18-s + 0.290·20-s − 2.90·21-s − 0.273·22-s − 1.77·23-s − 1.56·24-s + 25-s − 1.13·26-s + 3.43·27-s − 0.435·28-s + 2.20·30-s + ⋯
L(s)  = 1  + 1.13·2-s + 1.94·3-s + 0.290·4-s + 5-s + 2.20·6-s − 1.49·7-s − 0.805·8-s + 2.77·9-s + 1.13·10-s − 0.241·11-s + 0.564·12-s − 13-s − 1.70·14-s + 1.94·15-s − 1.20·16-s + 0.709·17-s + 3.14·18-s + 0.290·20-s − 2.90·21-s − 0.273·22-s − 1.77·23-s − 1.56·24-s + 25-s − 1.13·26-s + 3.43·27-s − 0.435·28-s + 2.20·30-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

\( d \)  =  \(2\)
\( N \)  =  \(2015\)    =    \(5 \cdot 13 \cdot 31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{2015} (2014, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2015,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(3.368946518\)
\(L(\frac12)\)  \(\approx\)  \(3.368946518\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;13,\;31\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{5,\;13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 - T \)
13 \( 1 + T \)
31 \( 1 + T \)
good2 \( 1 - 1.13T + T^{2} \)
3 \( 1 - 1.94T + T^{2} \)
7 \( 1 + 1.49T + T^{2} \)
11 \( 1 + 0.241T + T^{2} \)
17 \( 1 - 0.709T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.77T + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.13T + T^{2} \)
47 \( 1 - 1.77T + T^{2} \)
53 \( 1 + 0.241T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 0.241T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.49T + T^{2} \)
97 \( 1 + 1.94T + T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.390471966931875360828857037856, −8.768866554580884971153317059370, −7.73985492258028492306574763047, −6.92137697748881392507610985064, −6.13527699224682031543042169213, −5.22748661305882381447583102795, −4.12318974184031417557261487514, −3.43831153160476014823201189282, −2.74722524968468503534655783056, −2.07112127568161902052360501432, 2.07112127568161902052360501432, 2.74722524968468503534655783056, 3.43831153160476014823201189282, 4.12318974184031417557261487514, 5.22748661305882381447583102795, 6.13527699224682031543042169213, 6.92137697748881392507610985064, 7.73985492258028492306574763047, 8.768866554580884971153317059370, 9.390471966931875360828857037856

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