Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.25·2-s − 3-s − 0.429·4-s + 1.76·5-s + 1.25·6-s + 3.04·8-s + 9-s − 2.20·10-s − 4.84·11-s + 0.429·12-s + 3.18·13-s − 1.76·15-s − 2.95·16-s − 6.61·17-s − 1.25·18-s + 8.31·19-s − 0.757·20-s + 6.06·22-s − 8.32·23-s − 3.04·24-s − 1.89·25-s − 3.99·26-s − 27-s − 7.79·29-s + 2.20·30-s + 1.04·31-s − 2.38·32-s + ⋯
L(s)  = 1  − 0.886·2-s − 0.577·3-s − 0.214·4-s + 0.788·5-s + 0.511·6-s + 1.07·8-s + 0.333·9-s − 0.698·10-s − 1.46·11-s + 0.124·12-s + 0.884·13-s − 0.455·15-s − 0.739·16-s − 1.60·17-s − 0.295·18-s + 1.90·19-s − 0.169·20-s + 1.29·22-s − 1.73·23-s − 0.621·24-s − 0.378·25-s − 0.783·26-s − 0.192·27-s − 1.44·29-s + 0.403·30-s + 0.186·31-s − 0.421·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6027} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6524696254$
$L(\frac12)$  $\approx$  $0.6524696254$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 + 1.25T + 2T^{2} \)
5 \( 1 - 1.76T + 5T^{2} \)
11 \( 1 + 4.84T + 11T^{2} \)
13 \( 1 - 3.18T + 13T^{2} \)
17 \( 1 + 6.61T + 17T^{2} \)
19 \( 1 - 8.31T + 19T^{2} \)
23 \( 1 + 8.32T + 23T^{2} \)
29 \( 1 + 7.79T + 29T^{2} \)
31 \( 1 - 1.04T + 31T^{2} \)
37 \( 1 - 3.58T + 37T^{2} \)
43 \( 1 - 4.05T + 43T^{2} \)
47 \( 1 + 0.963T + 47T^{2} \)
53 \( 1 - 5.52T + 53T^{2} \)
59 \( 1 - 9.28T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 - 6.04T + 67T^{2} \)
71 \( 1 - 9.94T + 71T^{2} \)
73 \( 1 - 3.75T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + 8.11T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 - 11.2T + 97T^{2} \)
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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

−7.987392830416953162202210234871, −7.63182074636874573149814164114, −6.74181686716607211938610924382, −5.74033124069528436350142517421, −5.49797766979846078241049380850, −4.53563916925425359859529508129, −3.74914281105336977136744073097, −2.40732977924787423713999395662, −1.67860122454403620740246515539, −0.50457115338464414344420919581, 0.50457115338464414344420919581, 1.67860122454403620740246515539, 2.40732977924787423713999395662, 3.74914281105336977136744073097, 4.53563916925425359859529508129, 5.49797766979846078241049380850, 5.74033124069528436350142517421, 6.74181686716607211938610924382, 7.63182074636874573149814164114, 7.987392830416953162202210234871

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