Properties

Degree 2
Conductor $ 3 \cdot 5 $
Sign $1$
Motivic weight 7
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13·2-s − 27·3-s + 41·4-s − 125·5-s + 351·6-s + 1.38e3·7-s + 1.13e3·8-s + 729·9-s + 1.62e3·10-s − 3.30e3·11-s − 1.10e3·12-s + 8.50e3·13-s − 1.79e4·14-s + 3.37e3·15-s − 1.99e4·16-s − 9.99e3·17-s − 9.47e3·18-s + 4.12e4·19-s − 5.12e3·20-s − 3.72e4·21-s + 4.29e4·22-s + 8.41e4·23-s − 3.05e4·24-s + 1.56e4·25-s − 1.10e5·26-s − 1.96e4·27-s + 5.65e4·28-s + ⋯
L(s)  = 1  − 1.14·2-s − 0.577·3-s + 0.320·4-s − 0.447·5-s + 0.663·6-s + 1.52·7-s + 0.780·8-s + 1/3·9-s + 0.513·10-s − 0.748·11-s − 0.184·12-s + 1.07·13-s − 1.74·14-s + 0.258·15-s − 1.21·16-s − 0.493·17-s − 0.383·18-s + 1.37·19-s − 0.143·20-s − 0.877·21-s + 0.860·22-s + 1.44·23-s − 0.450·24-s + 1/5·25-s − 1.23·26-s − 0.192·27-s + 0.487·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15\)    =    \(3 \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  $\chi_{15} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 15,\ (\ :7/2),\ 1)$
$L(4)$  $\approx$  $0.715847$
$L(\frac12)$  $\approx$  $0.715847$
$L(\frac{9}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + p^{3} T \)
5 \( 1 + p^{3} T \)
good2 \( 1 + 13 T + p^{7} T^{2} \)
7 \( 1 - 1380 T + p^{7} T^{2} \)
11 \( 1 + 3304 T + p^{7} T^{2} \)
13 \( 1 - 8506 T + p^{7} T^{2} \)
17 \( 1 + 9994 T + p^{7} T^{2} \)
19 \( 1 - 41236 T + p^{7} T^{2} \)
23 \( 1 - 84120 T + p^{7} T^{2} \)
29 \( 1 - 132802 T + p^{7} T^{2} \)
31 \( 1 + 1800 p T + p^{7} T^{2} \)
37 \( 1 - 228170 T + p^{7} T^{2} \)
41 \( 1 + 139670 T + p^{7} T^{2} \)
43 \( 1 + 755492 T + p^{7} T^{2} \)
47 \( 1 - 836984 T + p^{7} T^{2} \)
53 \( 1 - 1641650 T + p^{7} T^{2} \)
59 \( 1 + 989656 T + p^{7} T^{2} \)
61 \( 1 + 1658162 T + p^{7} T^{2} \)
67 \( 1 + 4523844 T + p^{7} T^{2} \)
71 \( 1 + 389408 T + p^{7} T^{2} \)
73 \( 1 - 5617330 T + p^{7} T^{2} \)
79 \( 1 - 3901080 T + p^{7} T^{2} \)
83 \( 1 + 9394116 T + p^{7} T^{2} \)
89 \( 1 - 2803746 T + p^{7} T^{2} \)
97 \( 1 - 5099426 T + p^{7} T^{2} \)
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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

−18.02725727794992342875420889861, −16.74923279366807928809328352273, −15.45240049494764254447354444819, −13.59060322405581826033648118217, −11.50146321884975528724937612374, −10.62187303554169047678420609046, −8.698312502391348568212585851743, −7.51192222367118616194928232841, −4.89624300188903990329121188948, −1.08668704122281820100157322467, 1.08668704122281820100157322467, 4.89624300188903990329121188948, 7.51192222367118616194928232841, 8.698312502391348568212585851743, 10.62187303554169047678420609046, 11.50146321884975528724937612374, 13.59060322405581826033648118217, 15.45240049494764254447354444819, 16.74923279366807928809328352273, 18.02725727794992342875420889861

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