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 + 7.15·13-s − 8.19i·17-s + 4.35·19-s − 5·25-s + 7·49-s − 5.72i·53-s − 11.7i·59-s + 0.271i·71-s − 12.7·79-s + 14.8i·83-s − 17.7i·89-s + 13.2i·101-s − 17.4·109-s + 6.27i·113-s + ⋯
L(s)  = 1  − 1.00i·11-s + 1.98·13-s − 1.98i·17-s + 1.00·19-s − 25-s + 49-s − 0.786i·53-s − 1.52i·59-s + 0.0322i·71-s − 1.43·79-s + 1.62i·83-s − 1.87i·89-s + 1.32i·101-s − 1.67·109-s + 0.589i·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  =  $(2,\ 7524,\ (\ :1/2),\ i)$
$L(1)$  $\approx$  $2.057614035$
$L(\frac12)$  $\approx$  $2.057614035$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11,\;19\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;11,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + 3.31iT \)
19 \( 1 - 4.35T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 - 7T^{2} \)
13 \( 1 - 7.15T + 13T^{2} \)
17 \( 1 + 8.19iT - 17T^{2} \)
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 \( 1 + 5.72iT - 53T^{2} \)
59 \( 1 + 11.7iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 0.271iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 - 14.8iT - 83T^{2} \)
89 \( 1 + 17.7iT - 89T^{2} \)
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

−7.77214703020113197295966448849, −7.00744720735541037891381889252, −6.26967501886646491269629732450, −5.62581581852692223486781398493, −5.04967387412201591781055601922, −3.93156023178118080730542500792, −3.39588763653633242847046829135, −2.61945232739895613479722508011, −1.35613638680366418495357266129, −0.53747223767850330091223141646, 1.23331505670155620426558959304, 1.80100424765510532174729607984, 2.99525619500179013461861623699, 3.98763281262577800274074267653, 4.15664007103287164004883984157, 5.48241530768333575483698696158, 5.93164283733617122551840860438, 6.58766919177502822279200701284, 7.44675887170108060671024876556, 8.060590822556694165717505046373

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