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  =  $(2,\ 7524,\ (\ :1/2),\ -i)$
$L(1)$  $\approx$  $0.5074676483$
$L(\frac12)$  $\approx$  $0.5074676483$
$L(\frac{3}{2})$   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

−7.916398398972334653878316954348, −7.38405145193390019456276758844, −6.79873181681668687978814341652, −5.90822971973235372907942822747, −5.20514749607580846545961337502, −4.62725052403952637073409562087, −3.75038006020192462164685632467, −2.67688571323049050710653110020, −2.33795129951428205460981606912, −0.793693308409649463217673520114, 0.14694924778403813915291489496, 1.93017960363261647183386561467, 2.12940650971654575009921013153, 3.39657232461308309903623292875, 4.28015134027379989357709853376, 4.73454924857961062174490970101, 5.64394553341733118307269293040, 6.35467065869593966519331845934, 7.08319194477117176343317431110, 7.69134769375573005604325875534

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