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Properties

Label 2-3997-3997.3086-c0-0-0
Degree $2$
Conductor $3997$
Sign $0.0721 - 0.997i$
Analytic cond. $1.99476$
Root an. cond. $1.41236$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.268 + 0.0600i)2-s + (−0.836 − 0.394i)4-s + (−0.401 + 0.915i)7-s + (−0.417 − 0.324i)8-s + (0.716 − 0.697i)9-s + (−1.91 − 0.540i)11-s + (−0.162 + 0.221i)14-s + (0.495 + 0.601i)16-s + (0.234 − 0.143i)18-s + (−0.480 − 0.259i)22-s + (−0.471 + 0.366i)23-s + (0.451 + 0.892i)25-s + (0.697 − 0.607i)28-s + (0.114 + 0.585i)29-s + (0.335 + 0.662i)32-s + ⋯
L(s)  = 1  + (0.268 + 0.0600i)2-s + (−0.836 − 0.394i)4-s + (−0.401 + 0.915i)7-s + (−0.417 − 0.324i)8-s + (0.716 − 0.697i)9-s + (−1.91 − 0.540i)11-s + (−0.162 + 0.221i)14-s + (0.495 + 0.601i)16-s + (0.234 − 0.143i)18-s + (−0.480 − 0.259i)22-s + (−0.471 + 0.366i)23-s + (0.451 + 0.892i)25-s + (0.697 − 0.607i)28-s + (0.114 + 0.585i)29-s + (0.335 + 0.662i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0721 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0721 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $0.0721 - 0.997i$
Analytic conductor: \(1.99476\)
Root analytic conductor: \(1.41236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (3086, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3997,\ (\ :0),\ 0.0721 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6938518447\)
\(L(\frac12)\) \(\approx\) \(0.6938518447\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.401 - 0.915i)T \)
571 \( 1 + (-0.635 - 0.771i)T \)
good2 \( 1 + (-0.268 - 0.0600i)T + (0.904 + 0.426i)T^{2} \)
3 \( 1 + (-0.716 + 0.697i)T^{2} \)
5 \( 1 + (-0.451 - 0.892i)T^{2} \)
11 \( 1 + (1.91 + 0.540i)T + (0.851 + 0.523i)T^{2} \)
13 \( 1 + (-0.137 + 0.990i)T^{2} \)
17 \( 1 + (0.191 - 0.981i)T^{2} \)
19 \( 1 + (-0.716 + 0.697i)T^{2} \)
23 \( 1 + (0.471 - 0.366i)T + (0.245 - 0.969i)T^{2} \)
29 \( 1 + (-0.114 - 0.585i)T + (-0.926 + 0.376i)T^{2} \)
31 \( 1 + (-0.945 + 0.324i)T^{2} \)
37 \( 1 + (-1.32 - 1.15i)T + (0.137 + 0.990i)T^{2} \)
41 \( 1 + (-0.451 - 0.892i)T^{2} \)
43 \( 1 + (-1.39 - 1.36i)T + (0.0275 + 0.999i)T^{2} \)
47 \( 1 + (0.754 + 0.656i)T^{2} \)
53 \( 1 + (1.87 - 0.420i)T + (0.904 - 0.426i)T^{2} \)
59 \( 1 + (0.677 + 0.735i)T^{2} \)
61 \( 1 + (0.821 - 0.569i)T^{2} \)
67 \( 1 + (0.582 - 1.86i)T + (-0.821 - 0.569i)T^{2} \)
71 \( 1 + (0.245 - 0.425i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.926 + 0.376i)T^{2} \)
79 \( 1 + (0.225 + 0.156i)T + (0.350 + 0.936i)T^{2} \)
83 \( 1 + (0.754 + 0.656i)T^{2} \)
89 \( 1 + (0.191 - 0.981i)T^{2} \)
97 \( 1 + (-0.635 - 0.771i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.857078207556077809156420885327, −8.140671479247413894410467546786, −7.42409280100554731672939887290, −6.28710751740080219713979155965, −5.82239610331419032946842260288, −5.09288363986074860837390189836, −4.42185449323296823048987884917, −3.31057464467676258428387271527, −2.71717935331069583357117444648, −1.21168797047095637698440569444, 0.39277525323427880753263436294, 2.18443675148774866143810485800, 2.97872004061142613761337936100, 4.11337823066011407770815729332, 4.54334378823628343852350715329, 5.23867482581875035239077697217, 6.15673442127070914269020975195, 7.25549025590902361587445549796, 7.79783568340382962139233087417, 8.145718541634239425127285114943

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