Properties

Degree 2
Conductor $ 7 \cdot 571 $
Sign $0.893 - 0.449i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.341 − 1.74i)2-s + (−2.00 − 0.815i)4-s + (−0.461 + 0.887i)7-s + (−1.13 + 1.73i)8-s + (−0.739 − 0.673i)9-s + (−0.000667 + 0.121i)11-s + (1.39 + 1.10i)14-s + (1.09 + 1.06i)16-s + (−1.42 + 1.06i)18-s + (0.211 + 0.0425i)22-s + (−1.65 + 0.448i)23-s + (−0.795 − 0.605i)25-s + (1.64 − 1.40i)28-s + (−0.0415 + 1.50i)29-s + (0.527 − 0.366i)32-s + ⋯
L(s)  = 1  + (0.341 − 1.74i)2-s + (−2.00 − 0.815i)4-s + (−0.461 + 0.887i)7-s + (−1.13 + 1.73i)8-s + (−0.739 − 0.673i)9-s + (−0.000667 + 0.121i)11-s + (1.39 + 1.10i)14-s + (1.09 + 1.06i)16-s + (−1.42 + 1.06i)18-s + (0.211 + 0.0425i)22-s + (−1.65 + 0.448i)23-s + (−0.795 − 0.605i)25-s + (1.64 − 1.40i)28-s + (−0.0415 + 1.50i)29-s + (0.527 − 0.366i)32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3997\)    =    \(7 \cdot 571\)
\( \varepsilon \)  =  $0.893 - 0.449i$
motivic weight  =  \(0\)
character  :  $\chi_{3997} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3997,\ (\ :0),\ 0.893 - 0.449i)$
$L(\frac{1}{2})$  $\approx$  $0.2626400568$
$L(\frac12)$  $\approx$  $0.2626400568$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;571\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{7,\;571\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 + (0.461 - 0.887i)T \)
571 \( 1 + (-0.716 - 0.697i)T \)
good2 \( 1 + (-0.341 + 1.74i)T + (-0.926 - 0.376i)T^{2} \)
3 \( 1 + (0.739 + 0.673i)T^{2} \)
5 \( 1 + (0.795 + 0.605i)T^{2} \)
11 \( 1 + (0.000667 - 0.121i)T + (-0.999 - 0.0110i)T^{2} \)
13 \( 1 + (0.709 - 0.705i)T^{2} \)
17 \( 1 + (-0.959 - 0.282i)T^{2} \)
19 \( 1 + (-0.202 - 0.979i)T^{2} \)
23 \( 1 + (1.65 - 0.448i)T + (0.863 - 0.504i)T^{2} \)
29 \( 1 + (0.0415 - 1.50i)T + (-0.998 - 0.0550i)T^{2} \)
31 \( 1 + (0.879 + 0.475i)T^{2} \)
37 \( 1 + (0.977 + 0.0756i)T + (0.988 + 0.153i)T^{2} \)
41 \( 1 + (0.821 - 0.569i)T^{2} \)
43 \( 1 + (-0.719 - 1.59i)T + (-0.660 + 0.750i)T^{2} \)
47 \( 1 + (-0.851 + 0.523i)T^{2} \)
53 \( 1 + (-1.39 - 1.29i)T + (0.0715 + 0.997i)T^{2} \)
59 \( 1 + (-0.945 - 0.324i)T^{2} \)
61 \( 1 + (-0.471 + 0.882i)T^{2} \)
67 \( 1 + (0.0859 - 0.373i)T + (-0.899 - 0.436i)T^{2} \)
71 \( 1 + (1.46 - 0.310i)T + (0.913 - 0.406i)T^{2} \)
73 \( 1 + (0.360 + 0.932i)T^{2} \)
79 \( 1 + (1.77 + 0.863i)T + (0.618 + 0.785i)T^{2} \)
83 \( 1 + (0.997 + 0.0770i)T^{2} \)
89 \( 1 + (-0.565 + 0.824i)T^{2} \)
97 \( 1 + (0.441 - 0.897i)T^{2} \)
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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

−8.861971370786442764482451388291, −8.467554610396870925708266298247, −7.24472292611746704847163741306, −5.97828561494506181110701855794, −5.72797423871067861271862956364, −4.63292588103729675061794707504, −3.81195467122257448804390647889, −3.10380144748638864769154506783, −2.43251570347684529757939005045, −1.48306963084392906427555337055, 0.12638775975786622149061136615, 2.23200034575736678024829347672, 3.65142727525881319762397150474, 4.12772503492095030858790009492, 5.06773837747527141984417234991, 5.83845767935967839352254644828, 6.24167491747301118222323118584, 7.22179305422697188066141957974, 7.61404676851766301537321551860, 8.339441920676303994064748963875

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