Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.546·2-s + 3-s − 1.70·4-s − 5-s − 0.546·6-s − 7-s + 2.02·8-s + 9-s + 0.546·10-s − 11-s − 1.70·12-s + 4.56·13-s + 0.546·14-s − 15-s + 2.29·16-s + 2.70·17-s − 0.546·18-s − 8.02·19-s + 1.70·20-s − 21-s + 0.546·22-s − 3.52·23-s + 2.02·24-s + 25-s − 2.49·26-s + 27-s + 1.70·28-s + ⋯
L(s)  = 1  − 0.386·2-s + 0.577·3-s − 0.850·4-s − 0.447·5-s − 0.223·6-s − 0.377·7-s + 0.714·8-s + 0.333·9-s + 0.172·10-s − 0.301·11-s − 0.491·12-s + 1.26·13-s + 0.146·14-s − 0.258·15-s + 0.574·16-s + 0.655·17-s − 0.128·18-s − 1.84·19-s + 0.380·20-s − 0.218·21-s + 0.116·22-s − 0.735·23-s + 0.412·24-s + 0.200·25-s − 0.489·26-s + 0.192·27-s + 0.321·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1155} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1155,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.148602862\)
\(L(\frac12)\)  \(\approx\)  \(1.148602862\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + T \)
good2 \( 1 + 0.546T + 2T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 - 2.70T + 17T^{2} \)
19 \( 1 + 8.02T + 19T^{2} \)
23 \( 1 + 3.52T + 23T^{2} \)
29 \( 1 - 0.133T + 29T^{2} \)
31 \( 1 - 2.33T + 31T^{2} \)
37 \( 1 - 6.75T + 37T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 - 9.45T + 43T^{2} \)
47 \( 1 + 0.649T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 + 0.959T + 59T^{2} \)
61 \( 1 - 13.1T + 61T^{2} \)
67 \( 1 - 6.41T + 67T^{2} \)
71 \( 1 + 2.56T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 - 2.56T + 79T^{2} \)
83 \( 1 + 7.75T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 1.78T + 97T^{2} \)
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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

−9.651519638559645602827675118928, −8.879250463971963185401808986297, −8.242906332102032683302100200911, −7.72332197464850199867002268668, −6.50381514902619309030818384640, −5.58597614479635936194108424768, −4.17388398336443503037997852654, −3.88965084029950774652651027603, −2.46595098435056589530916554341, −0.854303664422338879810808974059, 0.854303664422338879810808974059, 2.46595098435056589530916554341, 3.88965084029950774652651027603, 4.17388398336443503037997852654, 5.58597614479635936194108424768, 6.50381514902619309030818384640, 7.72332197464850199867002268668, 8.242906332102032683302100200911, 8.879250463971963185401808986297, 9.651519638559645602827675118928

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