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Properties

Label 1-3997-3997.97-r1-0-0
Degree $1$
Conductor $3997$
Sign $0.997 + 0.0700i$
Analytic cond. $429.537$
Root an. cond. $429.537$
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  + (0.350 + 0.936i)2-s + (0.997 + 0.0770i)3-s + (−0.754 + 0.656i)4-s + (−0.411 − 0.911i)5-s + (0.277 + 0.960i)6-s + (−0.879 − 0.475i)8-s + (0.988 + 0.153i)9-s + (0.709 − 0.705i)10-s + (−0.256 + 0.966i)11-s + (−0.802 + 0.596i)12-s + (0.441 − 0.897i)13-s + (−0.340 − 0.940i)15-s + (0.137 − 0.990i)16-s + (0.899 + 0.436i)17-s + (0.202 + 0.979i)18-s + (−0.761 − 0.648i)19-s + ⋯
L(s)  = 1  + (0.350 + 0.936i)2-s + (0.997 + 0.0770i)3-s + (−0.754 + 0.656i)4-s + (−0.411 − 0.911i)5-s + (0.277 + 0.960i)6-s + (−0.879 − 0.475i)8-s + (0.988 + 0.153i)9-s + (0.709 − 0.705i)10-s + (−0.256 + 0.966i)11-s + (−0.802 + 0.596i)12-s + (0.441 − 0.897i)13-s + (−0.340 − 0.940i)15-s + (0.137 − 0.990i)16-s + (0.899 + 0.436i)17-s + (0.202 + 0.979i)18-s + (−0.761 − 0.648i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $0.997 + 0.0700i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ 0.997 + 0.0700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.114603430 + 0.1092000238i\)
\(L(\frac12)\) \(\approx\) \(3.114603430 + 0.1092000238i\)
\(L(1)\) \(\approx\) \(1.426189661 + 0.5114195451i\)
\(L(1)\) \(\approx\) \(1.426189661 + 0.5114195451i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (0.350 + 0.936i)T \)
3 \( 1 + (0.997 + 0.0770i)T \)
5 \( 1 + (-0.411 - 0.911i)T \)
11 \( 1 + (-0.256 + 0.966i)T \)
13 \( 1 + (0.441 - 0.897i)T \)
17 \( 1 + (0.899 + 0.436i)T \)
19 \( 1 + (-0.761 - 0.648i)T \)
23 \( 1 + (0.991 + 0.131i)T \)
29 \( 1 + (-0.962 + 0.272i)T \)
31 \( 1 + (-0.789 - 0.614i)T \)
37 \( 1 + (0.884 + 0.466i)T \)
41 \( 1 + (-0.993 + 0.110i)T \)
43 \( 1 + (-0.889 - 0.456i)T \)
47 \( 1 + (0.926 - 0.376i)T \)
53 \( 1 + (-0.782 - 0.622i)T \)
59 \( 1 + (0.0825 + 0.996i)T \)
61 \( 1 + (0.968 - 0.250i)T \)
67 \( 1 + (0.930 - 0.366i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (0.0385 + 0.999i)T \)
79 \( 1 + (0.731 - 0.681i)T \)
83 \( 1 + (-0.970 - 0.240i)T \)
89 \( 1 + (0.693 - 0.720i)T \)
97 \( 1 + (0.857 + 0.514i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.70147285117862932976333206874, −18.21236802588208545216712476380, −16.91590395168163348985817048111, −16.12923079912381266301943026745, −15.26255629641227911885938322499, −14.636853998415597372702947049992, −14.18422567066450619453500390042, −13.630382267624424165152300098802, −12.86508613640706372645119219306, −12.196921419830196708872534499427, −11.177116869455319113774342877179, −10.98122277998231778214089113129, −10.01683344665146750869588826877, −9.42107770715433232900771069470, −8.6078525416708958798852893693, −8.04452144774495050639086023906, −7.10984174016158471084474762671, −6.34280159567882986423634641696, −5.458719112241944529218978905434, −4.40408323601246787655251319488, −3.602393432073312537989520112146, −3.320178661774372901276305226443, −2.470240346038034948906237852655, −1.73912575576576288216613788040, −0.77918291579023819788027482947, 0.42356780048249490311933427220, 1.459224457420255330677316785255, 2.546151536466780569866817294783, 3.589247275887504474991668526448, 3.938596857342355001175112469324, 5.00141454638351971269730220815, 5.26394348350248249924425403968, 6.47318808914128759525363438144, 7.33881017056090309742991130797, 7.83485900730613177428143290198, 8.41892696556930739268068283687, 9.075502715494743068599749087303, 9.67712666828609445601072478296, 10.53022921109094031482602380818, 11.71284949919224506291044959775, 12.64354813041648207393673961634, 13.058395718964335193776635114290, 13.32253802310621514851852829232, 14.53150964207592678200355900335, 15.04980470323793672494288425914, 15.3818233401545616811499766, 16.10314228464383521118040883974, 16.960349785358981457728712045061, 17.30351867925048478803615917088, 18.47763718157715431474305600576

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