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Properties

Label 2-3997-3997.1224-c0-0-0
Degree $2$
Conductor $3997$
Sign $-0.973 + 0.228i$
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  + (−1.87 − 0.531i)2-s + (2.39 + 1.47i)4-s + (0.789 − 0.614i)7-s + (−2.39 − 2.59i)8-s + (−0.821 − 0.569i)9-s + (0.248 − 0.117i)11-s + (−1.80 + 0.734i)14-s + (1.84 + 3.64i)16-s + (1.24 + 1.50i)18-s + (−0.528 + 0.0882i)22-s + (−0.970 + 1.05i)23-s + (−0.191 − 0.981i)25-s + (2.79 − 0.308i)28-s + (−0.849 − 1.15i)29-s + (−0.849 − 4.35i)32-s + ⋯
L(s)  = 1  + (−1.87 − 0.531i)2-s + (2.39 + 1.47i)4-s + (0.789 − 0.614i)7-s + (−2.39 − 2.59i)8-s + (−0.821 − 0.569i)9-s + (0.248 − 0.117i)11-s + (−1.80 + 0.734i)14-s + (1.84 + 3.64i)16-s + (1.24 + 1.50i)18-s + (−0.528 + 0.0882i)22-s + (−0.970 + 1.05i)23-s + (−0.191 − 0.981i)25-s + (2.79 − 0.308i)28-s + (−0.849 − 1.15i)29-s + (−0.849 − 4.35i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.973 + 0.228i$
Analytic conductor: \(1.99476\)
Root analytic conductor: \(1.41236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (1224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3997,\ (\ :0),\ -0.973 + 0.228i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2900273471\)
\(L(\frac12)\) \(\approx\) \(0.2900273471\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.789 + 0.614i)T \)
571 \( 1 + (-0.451 - 0.892i)T \)
good2 \( 1 + (1.87 + 0.531i)T + (0.851 + 0.523i)T^{2} \)
3 \( 1 + (0.821 + 0.569i)T^{2} \)
5 \( 1 + (0.191 + 0.981i)T^{2} \)
11 \( 1 + (-0.248 + 0.117i)T + (0.635 - 0.771i)T^{2} \)
13 \( 1 + (-0.975 + 0.218i)T^{2} \)
17 \( 1 + (0.592 - 0.805i)T^{2} \)
19 \( 1 + (0.821 + 0.569i)T^{2} \)
23 \( 1 + (0.970 - 1.05i)T + (-0.0825 - 0.996i)T^{2} \)
29 \( 1 + (0.849 + 1.15i)T + (-0.298 + 0.954i)T^{2} \)
31 \( 1 + (0.401 + 0.915i)T^{2} \)
37 \( 1 + (1.96 + 0.217i)T + (0.975 + 0.218i)T^{2} \)
41 \( 1 + (0.191 + 0.981i)T^{2} \)
43 \( 1 + (-1.58 + 1.09i)T + (0.350 - 0.936i)T^{2} \)
47 \( 1 + (-0.993 - 0.110i)T^{2} \)
53 \( 1 + (1.74 - 0.492i)T + (0.851 - 0.523i)T^{2} \)
59 \( 1 + (-0.245 + 0.969i)T^{2} \)
61 \( 1 + (-0.0275 + 0.999i)T^{2} \)
67 \( 1 + (1.37 + 1.34i)T + (0.0275 + 0.999i)T^{2} \)
71 \( 1 + (-0.0825 - 0.143i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.298 + 0.954i)T^{2} \)
79 \( 1 + (-0.0537 - 1.95i)T + (-0.998 + 0.0550i)T^{2} \)
83 \( 1 + (-0.993 - 0.110i)T^{2} \)
89 \( 1 + (0.592 - 0.805i)T^{2} \)
97 \( 1 + (-0.451 - 0.892i)T^{2} \)
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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

−8.350331597356849569285450197789, −7.81695120247312586823408860688, −7.23408101710966945379968476231, −6.35209485000246257231438261936, −5.64227799812164964730040607508, −4.07799999640305755636943915926, −3.40441390321645816340949248215, −2.30347682498564940357784910032, −1.52495059698150340613469126467, −0.29250327536497462673969260910, 1.50293070992114698933346836528, 2.14378913929116217569169569271, 3.12553271194469155598922792565, 4.83106846295012236477802593701, 5.61186348114186674119770162525, 6.15661739654688580890787233096, 7.08535915267466772831339252559, 7.76983107362728296557736742838, 8.250794140840652905009466390197, 9.010218024300529992660848958974

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