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  :  $\chi_{3997} (615, \cdot )$
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 $p \notin \{7,\;571\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{7,\;571\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
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

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

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