Properties

Degree 2
Conductor $ 2^{7} \cdot 19 $
Sign $-i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·3-s + 5-s − 7-s − 1.00·9-s + 11-s + 1.41i·15-s + 17-s + 19-s − 1.41i·21-s − 1.41i·29-s + 1.41i·31-s + 1.41i·33-s − 35-s + 1.41i·37-s + 1.41i·41-s + ⋯
L(s)  = 1  + 1.41i·3-s + 5-s − 7-s − 1.00·9-s + 11-s + 1.41i·15-s + 17-s + 19-s − 1.41i·21-s − 1.41i·29-s + 1.41i·31-s + 1.41i·33-s − 35-s + 1.41i·37-s + 1.41i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2432\)    =    \(2^{7} \cdot 19\)
\( \varepsilon \)  =  $-i$
motivic weight  =  \(0\)
character  :  $\chi_{2432} (1025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2432,\ (\ :0),\ -i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.394108835\)
\(L(\frac12)\)  \(\approx\)  \(1.394108835\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;19\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 - T \)
good3 \( 1 - 1.41iT - T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + 1.41iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + T^{2} \)
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

−9.684980502553304155216705643809, −8.922507338867491402626439943921, −7.968873330144470489835380398063, −6.70288708059089294993608278330, −6.19056472651110000227843988914, −5.31217614313115425253310201074, −4.62759204067765548397395609711, −3.46400379384153039385833720929, −3.13738320677443425497528763966, −1.53318185159870885012074061222, 1.06602819575941458445746130751, 1.93331361085573211956091496664, 2.96023488152114112165797315315, 3.89527742979712287928193362844, 5.52831608313672084366181938018, 5.86054244901229944584324310744, 6.78974174744660007977565909557, 7.13590312465581883611692345281, 8.037016542285944529528452076250, 9.046714716080296934919855591226

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