Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.23·2-s + 3·3-s + 9.94·4-s − 5·5-s − 12.7·6-s − 7·7-s − 8.23·8-s + 9·9-s + 21.1·10-s − 41.5·11-s + 29.8·12-s + 88.9·13-s + 29.6·14-s − 15·15-s − 44.6·16-s − 120.·17-s − 38.1·18-s − 112.·19-s − 49.7·20-s − 21·21-s + 175.·22-s − 115.·23-s − 24.7·24-s + 25·25-s − 376.·26-s + 27·27-s − 69.6·28-s + ⋯
L(s)  = 1  − 1.49·2-s + 0.577·3-s + 1.24·4-s − 0.447·5-s − 0.864·6-s − 0.377·7-s − 0.363·8-s + 0.333·9-s + 0.669·10-s − 1.13·11-s + 0.717·12-s + 1.89·13-s + 0.566·14-s − 0.258·15-s − 0.697·16-s − 1.71·17-s − 0.499·18-s − 1.35·19-s − 0.555·20-s − 0.218·21-s + 1.70·22-s − 1.04·23-s − 0.210·24-s + 0.200·25-s − 2.84·26-s + 0.192·27-s − 0.469·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

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

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - 3T \)
5 \( 1 + 5T \)
7 \( 1 + 7T \)
good2 \( 1 + 4.23T + 8T^{2} \)
11 \( 1 + 41.5T + 1.33e3T^{2} \)
13 \( 1 - 88.9T + 2.19e3T^{2} \)
17 \( 1 + 120.T + 4.91e3T^{2} \)
19 \( 1 + 112.T + 6.85e3T^{2} \)
23 \( 1 + 115.T + 1.21e4T^{2} \)
29 \( 1 + 144.T + 2.43e4T^{2} \)
31 \( 1 + 258.T + 2.97e4T^{2} \)
37 \( 1 - 48.3T + 5.06e4T^{2} \)
41 \( 1 - 200.T + 6.89e4T^{2} \)
43 \( 1 + 218.T + 7.95e4T^{2} \)
47 \( 1 - 575.T + 1.03e5T^{2} \)
53 \( 1 + 184.T + 1.48e5T^{2} \)
59 \( 1 + 151.T + 2.05e5T^{2} \)
61 \( 1 + 529.T + 2.26e5T^{2} \)
67 \( 1 - 1.28T + 3.00e5T^{2} \)
71 \( 1 + 61.4T + 3.57e5T^{2} \)
73 \( 1 - 484.T + 3.89e5T^{2} \)
79 \( 1 - 878.T + 4.93e5T^{2} \)
83 \( 1 - 491.T + 5.71e5T^{2} \)
89 \( 1 + 415.T + 7.04e5T^{2} \)
97 \( 1 + 1.03e3T + 9.12e5T^{2} \)
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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

−12.88181569707492296853043281622, −11.02195019693420594559702105497, −10.65386272950849421210188056087, −9.168981172629181272824934474395, −8.535523732055619904725233737761, −7.62799052378416580530155802593, −6.33003777724750842570572638412, −4.01589893874121749634155343159, −2.09774235159822270881569792749, 0, 2.09774235159822270881569792749, 4.01589893874121749634155343159, 6.33003777724750842570572638412, 7.62799052378416580530155802593, 8.535523732055619904725233737761, 9.168981172629181272824934474395, 10.65386272950849421210188056087, 11.02195019693420594559702105497, 12.88181569707492296853043281622

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