Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 11 \cdot 19 $
Sign $-i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.31i·11-s − 2.79·13-s − 4.88i·17-s + 4.35·19-s − 5·25-s + 7·49-s + 8.72i·53-s + 2.72i·59-s + 14.7i·71-s + 17.1·79-s − 1.75i·83-s − 3.27i·89-s − 13.2i·101-s − 17.4·109-s + 20.7i·113-s + ⋯
L(s)  = 1  + 1.00i·11-s − 0.775·13-s − 1.18i·17-s + 1.00·19-s − 25-s + 49-s + 1.19i·53-s + 0.355i·59-s + 1.74i·71-s + 1.92·79-s − 0.192i·83-s − 0.346i·89-s − 1.32i·101-s − 1.67·109-s + 1.94i·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7524\)    =    \(2^{2} \cdot 3^{2} \cdot 11 \cdot 19\)
\( \varepsilon \)  =  $-i$
motivic weight  =  \(1\)
character  :  $\chi_{7524} (2089, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 7524,\ (\ :1/2),\ -i)\)
\(L(1)\)  \(\approx\)  \(1.320342450\)
\(L(\frac12)\)  \(\approx\)  \(1.320342450\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;19\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 - 3.31iT \)
19 \( 1 - 4.35T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 - 7T^{2} \)
13 \( 1 + 2.79T + 13T^{2} \)
17 \( 1 + 4.88iT - 17T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 8.72iT - 53T^{2} \)
59 \( 1 - 2.72iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 14.7iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 + 1.75iT - 83T^{2} \)
89 \( 1 + 3.27iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−7.80844506687874279226971656608, −7.38390338720132679826217221268, −6.88966036490446343580110387190, −5.88642044267051179172700670178, −5.18871050149975344473231467998, −4.60415017282693730568558099230, −3.78729339707179229432266287478, −2.78408458280605220181841940106, −2.14641450627228403128946268609, −0.975413134280614393802005143722, 0.35544010580295572867287398026, 1.54080482326575401819105250747, 2.48852958374385687561246549992, 3.42695994872383856084380387440, 3.97867928390240849805444290673, 5.01878147752781845487843716308, 5.61217211956235493564289530179, 6.28212979488752551640863389536, 7.00528335766621916188209368420, 7.932357598514393710540418701146

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