Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-0.894 + 0.447i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.38i·3-s − 2.35i·5-s − 0.244i·7-s + 1.07·9-s − 6.06i·11-s + 0.379·13-s − 3.26·15-s + (3.68 − 1.84i)17-s + 1.83·19-s − 0.340·21-s − 3.35i·23-s − 0.529·25-s − 5.65i·27-s + 0.959i·29-s + 7.69i·31-s + ⋯
L(s)  = 1  − 0.801i·3-s − 1.05i·5-s − 0.0925i·7-s + 0.356·9-s − 1.82i·11-s + 0.105·13-s − 0.843·15-s + (0.894 − 0.447i)17-s + 0.421·19-s − 0.0742·21-s − 0.698i·23-s − 0.105·25-s − 1.08i·27-s + 0.178i·29-s + 1.38i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-0.894 + 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -0.894 + 0.447i)$
$L(1)$  $\approx$  $2.148946836$
$L(\frac12)$  $\approx$  $2.148946836$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;17,\;59\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;17,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-3.68 + 1.84i)T \)
59 \( 1 - T \)
good3 \( 1 + 1.38iT - 3T^{2} \)
5 \( 1 + 2.35iT - 5T^{2} \)
7 \( 1 + 0.244iT - 7T^{2} \)
11 \( 1 + 6.06iT - 11T^{2} \)
13 \( 1 - 0.379T + 13T^{2} \)
19 \( 1 - 1.83T + 19T^{2} \)
23 \( 1 + 3.35iT - 23T^{2} \)
29 \( 1 - 0.959iT - 29T^{2} \)
31 \( 1 - 7.69iT - 31T^{2} \)
37 \( 1 + 5.21iT - 37T^{2} \)
41 \( 1 + 9.98iT - 41T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 - 2.89T + 47T^{2} \)
53 \( 1 - 9.23T + 53T^{2} \)
61 \( 1 - 5.34iT - 61T^{2} \)
67 \( 1 + 3.05T + 67T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 + 4.32iT - 73T^{2} \)
79 \( 1 - 9.09iT - 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 - 17.9iT - 97T^{2} \)
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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

−8.262070158440696202398720354887, −7.36754337969468121035317267528, −6.81982128173949776153150975295, −5.68809481001583146725844290391, −5.45581087256489422159242991194, −4.30758933639440734630168271045, −3.47778988121735950091458666752, −2.45778950676404056870163241346, −1.10205954181205618684296318730, −0.75891396391338181385229270902, 1.48847242975869610281500577150, 2.51954924023506799004195027800, 3.43994834180172721874257288258, 4.19626197968888650094523074371, 4.86145198908440318818041320426, 5.78067327899456832243919835086, 6.62798240037284156261429847085, 7.43491607965883176875844303504, 7.70279624847697895305723713280, 8.972563684971122936771339036062

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