Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 8·4-s + 5·5-s + 7·7-s + 9·9-s + 42·11-s + 24·12-s + 20·13-s − 15·15-s + 64·16-s + 66·17-s + 38·19-s − 40·20-s − 21·21-s + 12·23-s + 25·25-s − 27·27-s − 56·28-s − 258·29-s + 146·31-s − 126·33-s + 35·35-s − 72·36-s + 434·37-s − 60·39-s − 282·41-s + 20·43-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.15·11-s + 0.577·12-s + 0.426·13-s − 0.258·15-s + 16-s + 0.941·17-s + 0.458·19-s − 0.447·20-s − 0.218·21-s + 0.108·23-s + 1/5·25-s − 0.192·27-s − 0.377·28-s − 1.65·29-s + 0.845·31-s − 0.664·33-s + 0.169·35-s − 1/3·36-s + 1.92·37-s − 0.246·39-s − 1.07·41-s + 0.0709·43-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  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(1.26100\)
\(L(\frac12)\)  \(\approx\)  \(1.26100\)
\(L(\frac{5}{2})\)   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 \( 1 + p T \)
5 \( 1 - p T \)
7 \( 1 - p T \)
good2 \( 1 + p^{3} T^{2} \)
11 \( 1 - 42 T + p^{3} T^{2} \)
13 \( 1 - 20 T + p^{3} T^{2} \)
17 \( 1 - 66 T + p^{3} T^{2} \)
19 \( 1 - 2 p T + p^{3} T^{2} \)
23 \( 1 - 12 T + p^{3} T^{2} \)
29 \( 1 + 258 T + p^{3} T^{2} \)
31 \( 1 - 146 T + p^{3} T^{2} \)
37 \( 1 - 434 T + p^{3} T^{2} \)
41 \( 1 + 282 T + p^{3} T^{2} \)
43 \( 1 - 20 T + p^{3} T^{2} \)
47 \( 1 + 72 T + p^{3} T^{2} \)
53 \( 1 - 336 T + p^{3} T^{2} \)
59 \( 1 + 360 T + p^{3} T^{2} \)
61 \( 1 + 682 T + p^{3} T^{2} \)
67 \( 1 - 812 T + p^{3} T^{2} \)
71 \( 1 - 810 T + p^{3} T^{2} \)
73 \( 1 + 124 T + p^{3} T^{2} \)
79 \( 1 - 1136 T + p^{3} T^{2} \)
83 \( 1 - 156 T + p^{3} T^{2} \)
89 \( 1 + 1038 T + p^{3} T^{2} \)
97 \( 1 - 1208 T + p^{3} T^{2} \)
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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

−13.33999715747491387220601626537, −12.25507757611512237536921989041, −11.24148326992387271744824846264, −9.917697286121603683278694291785, −9.140012825049310871337886399587, −7.81787339196846720357315904867, −6.21817264354172821605570724071, −5.13556761134837335702729587063, −3.81319902620835772775865421004, −1.14789243311335868174781089344, 1.14789243311335868174781089344, 3.81319902620835772775865421004, 5.13556761134837335702729587063, 6.21817264354172821605570724071, 7.81787339196846720357315904867, 9.140012825049310871337886399587, 9.917697286121603683278694291785, 11.24148326992387271744824846264, 12.25507757611512237536921989041, 13.33999715747491387220601626537

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