Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $0.540 - 0.841i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.43i·3-s − 2.66i·5-s − 3.93i·7-s + 0.931·9-s + 4.93i·11-s + 5.59·13-s + 3.83·15-s + (−2.22 + 3.46i)17-s − 6.57·19-s + 5.65·21-s + 6.13i·23-s − 2.09·25-s + 5.65i·27-s − 1.64i·29-s + 9.54i·31-s + ⋯
L(s)  = 1  + 0.830i·3-s − 1.19i·5-s − 1.48i·7-s + 0.310·9-s + 1.48i·11-s + 1.55·13-s + 0.989·15-s + (−0.540 + 0.841i)17-s − 1.50·19-s + 1.23·21-s + 1.27i·23-s − 0.419·25-s + 1.08i·27-s − 0.305i·29-s + 1.71i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $0.540 - 0.841i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ 0.540 - 0.841i)$
$L(1)$  $\approx$  $1.851538807$
$L(\frac12)$  $\approx$  $1.851538807$
$L(\frac{3}{2})$   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 \( 1 + (2.22 - 3.46i)T \)
59 \( 1 - T \)
good3 \( 1 - 1.43iT - 3T^{2} \)
5 \( 1 + 2.66iT - 5T^{2} \)
7 \( 1 + 3.93iT - 7T^{2} \)
11 \( 1 - 4.93iT - 11T^{2} \)
13 \( 1 - 5.59T + 13T^{2} \)
19 \( 1 + 6.57T + 19T^{2} \)
23 \( 1 - 6.13iT - 23T^{2} \)
29 \( 1 + 1.64iT - 29T^{2} \)
31 \( 1 - 9.54iT - 31T^{2} \)
37 \( 1 - 6.71iT - 37T^{2} \)
41 \( 1 + 9.66iT - 41T^{2} \)
43 \( 1 - 9.17T + 43T^{2} \)
47 \( 1 + 8.93T + 47T^{2} \)
53 \( 1 - 5.74T + 53T^{2} \)
61 \( 1 - 11.2iT - 61T^{2} \)
67 \( 1 + 0.829T + 67T^{2} \)
71 \( 1 + 5.67iT - 71T^{2} \)
73 \( 1 - 5.67iT - 73T^{2} \)
79 \( 1 - 2.84iT - 79T^{2} \)
83 \( 1 - 4.48T + 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 - 11.1iT - 97T^{2} \)
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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

−8.735511361645385640419869485345, −7.926822967100887528383557814895, −7.07222217186515507847328954376, −6.46443993358210116711521304501, −5.31159197486269196899201442822, −4.53996043804791996490172651090, −4.11480005800424166288599779554, −3.64169950341127674793083151476, −1.77372979100999697955559990636, −1.14768671355978229799612427005, 0.58613791287305405045462449076, 2.08115857851838001408306180124, 2.59803337712110681741044237332, 3.47146441326232438587985456208, 4.48840017562596712683345114170, 5.84009877258909887830763272008, 6.27060479668364456758415146840, 6.54235717324075734062442169522, 7.61123261861059445540842418334, 8.456253022865041215188267005137

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