Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.23·2-s − 3-s + 3.00·4-s − 5-s + 2.23·6-s + 7-s − 2.23·8-s + 9-s + 2.23·10-s + 6.47·11-s − 3.00·12-s + 4.47·13-s − 2.23·14-s + 15-s − 0.999·16-s − 2·17-s − 2.23·18-s − 2.47·19-s − 3.00·20-s − 21-s − 14.4·22-s + 4·23-s + 2.23·24-s + 25-s − 10.0·26-s − 27-s + 3.00·28-s + ⋯
L(s)  = 1  − 1.58·2-s − 0.577·3-s + 1.50·4-s − 0.447·5-s + 0.912·6-s + 0.377·7-s − 0.790·8-s + 0.333·9-s + 0.707·10-s + 1.95·11-s − 0.866·12-s + 1.24·13-s − 0.597·14-s + 0.258·15-s − 0.249·16-s − 0.485·17-s − 0.527·18-s − 0.567·19-s − 0.670·20-s − 0.218·21-s − 3.08·22-s + 0.834·23-s + 0.456·24-s + 0.200·25-s − 1.96·26-s − 0.192·27-s + 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{105} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.453480$
$L(\frac12)$  $\approx$  $0.453480$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
good2 \( 1 + 2.23T + 2T^{2} \)
11 \( 1 - 6.47T + 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 2.47T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 1.52T + 31T^{2} \)
37 \( 1 + 6.94T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 8.94T + 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 + 3.52T + 53T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 5.52T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 8.47T + 97T^{2} \)
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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

−13.83105489238817075034959756976, −12.21462615368365629827364694348, −11.25571416032418356121748741329, −10.71248176543795167070725589629, −9.200136912201529736656132372640, −8.627168896087875885705621690209, −7.22411327079707474917124684286, −6.28297263652039351876956159723, −4.14755026085407614211409726178, −1.31435688386558938615415944577, 1.31435688386558938615415944577, 4.14755026085407614211409726178, 6.28297263652039351876956159723, 7.22411327079707474917124684286, 8.627168896087875885705621690209, 9.200136912201529736656132372640, 10.71248176543795167070725589629, 11.25571416032418356121748741329, 12.21462615368365629827364694348, 13.83105489238817075034959756976

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