Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 17 $
Sign $-0.816 + 0.577i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 1.41i·7-s − 11-s − 13-s − 17-s − 19-s + 23-s − 1.41i·31-s + 1.41i·35-s + 1.41i·37-s − 41-s + 43-s − 1.41i·47-s − 1.00·49-s − 1.41i·53-s + ⋯
L(s)  = 1  − 5-s − 1.41i·7-s − 11-s − 13-s − 17-s − 19-s + 23-s − 1.41i·31-s + 1.41i·35-s + 1.41i·37-s − 41-s + 43-s − 1.41i·47-s − 1.00·49-s − 1.41i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
\( \varepsilon \)  =  $-0.816 + 0.577i$
motivic weight  =  \(0\)
character  :  $\chi_{1224} (305, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1224,\ (\ :0),\ -0.816 + 0.577i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.3823805483\)
\(L(\frac12)\)  \(\approx\)  \(0.3823805483\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;17\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + T + T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.41iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.41iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.798387357837920282483225843388, −8.571124003580642402683794729419, −7.86793445908189252636694017840, −7.22604201157980910646828305987, −6.55652904619864707357625669315, −5.05062291657153624056412159918, −4.39834475904926607730509203379, −3.55991240279185510432456386374, −2.29341822991056717597436260083, −0.30799226672292309201947228854, 2.24228587948386791577121929546, 2.98246259987145635389225049074, 4.36632991453827353164468195972, 5.09168462835072136516363774183, 6.01753732011591840324226853568, 7.11853972412115120464203667236, 7.80028273994966925569329579545, 8.762271147200267393356025636917, 9.108580759732845590274149584498, 10.39767306844368612725729412196

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