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Properties

Label 1-3997-3997.719-r1-0-0
Degree $1$
Conductor $3997$
Sign $-0.628 - 0.777i$
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  + (−0.298 + 0.954i)2-s + (−0.471 + 0.882i)3-s + (−0.821 − 0.569i)4-s + (0.942 − 0.335i)5-s + (−0.701 − 0.712i)6-s + (0.789 − 0.614i)8-s + (−0.556 − 0.831i)9-s + (0.0385 + 0.999i)10-s + (0.884 − 0.466i)11-s + (0.889 − 0.456i)12-s + (−0.601 − 0.799i)13-s + (−0.148 + 0.988i)15-s + (0.350 + 0.936i)16-s + (0.537 + 0.843i)17-s + (0.959 − 0.282i)18-s + (−0.0715 + 0.997i)19-s + ⋯
L(s)  = 1  + (−0.298 + 0.954i)2-s + (−0.471 + 0.882i)3-s + (−0.821 − 0.569i)4-s + (0.942 − 0.335i)5-s + (−0.701 − 0.712i)6-s + (0.789 − 0.614i)8-s + (−0.556 − 0.831i)9-s + (0.0385 + 0.999i)10-s + (0.884 − 0.466i)11-s + (0.889 − 0.456i)12-s + (−0.601 − 0.799i)13-s + (−0.148 + 0.988i)15-s + (0.350 + 0.936i)16-s + (0.537 + 0.843i)17-s + (0.959 − 0.282i)18-s + (−0.0715 + 0.997i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.628 - 0.777i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ -0.628 - 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2280745383 + 0.4777984847i\)
\(L(\frac12)\) \(\approx\) \(-0.2280745383 + 0.4777984847i\)
\(L(1)\) \(\approx\) \(0.6640736895 + 0.4599268101i\)
\(L(1)\) \(\approx\) \(0.6640736895 + 0.4599268101i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (-0.298 + 0.954i)T \)
3 \( 1 + (-0.471 + 0.882i)T \)
5 \( 1 + (0.942 - 0.335i)T \)
11 \( 1 + (0.884 - 0.466i)T \)
13 \( 1 + (-0.601 - 0.799i)T \)
17 \( 1 + (0.537 + 0.843i)T \)
19 \( 1 + (-0.0715 + 0.997i)T \)
23 \( 1 + (-0.693 + 0.720i)T \)
29 \( 1 + (0.945 - 0.324i)T \)
31 \( 1 + (-0.635 + 0.771i)T \)
37 \( 1 + (0.874 - 0.485i)T \)
41 \( 1 + (0.879 + 0.475i)T \)
43 \( 1 + (-0.768 - 0.639i)T \)
47 \( 1 + (-0.350 - 0.936i)T \)
53 \( 1 + (-0.999 - 0.0110i)T \)
59 \( 1 + (-0.993 - 0.110i)T \)
61 \( 1 + (-0.802 + 0.596i)T \)
67 \( 1 + (-0.999 - 0.0110i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.857 + 0.514i)T \)
79 \( 1 + (0.999 + 0.0220i)T \)
83 \( 1 + (-0.701 - 0.712i)T \)
89 \( 1 + (-0.266 + 0.963i)T \)
97 \( 1 + (0.973 - 0.229i)T \)
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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

−17.95744933546594043137652078756, −17.46237884584850964578798717637, −16.81464000107480260610939545983, −16.31061655849717577320089514728, −14.75203095616998584031425048479, −14.13975782332225612370461294267, −13.757231146254822117010383679514, −12.88973613930280653808363161067, −12.33704142988474650656708163792, −11.68419639182092857345075875955, −11.11883386172490396169444395876, −10.3538880707897318299386441584, −9.402354146642753639098517159358, −9.28583712062467843907818451034, −8.0959529438393023485603066506, −7.32098840050558049082329820992, −6.65151235270914288506068223394, −5.97626230226759792755191865922, −4.858052501636698098046921327602, −4.4737240218928172407430906601, −3.03729245496486911501915452733, −2.47921079511787672642548788519, −1.74795662880579640235778876213, −1.097453465328517706535362784725, −0.10661826867669406715390527084, 0.94319431021682563314237268481, 1.698967742444341806279091409668, 3.18604872912271345855938259817, 3.96076621916382746398839764041, 4.77109924198387726026575190754, 5.5190072773042099218519214353, 6.00692578042167716385178805635, 6.45838802911921081658463347699, 7.662072349588682460772753576216, 8.387338835038356566294471215129, 9.10065742042340066723440704425, 9.73762202092891426696703793999, 10.226623523690240427480010472852, 10.80250328580245637167887565351, 12.01299384742649911333999904635, 12.576850437067086167332748607922, 13.5255313093754896718329507234, 14.34941181783729291589809824491, 14.623289042923194684821288607436, 15.438323214333154318932189549231, 16.25507392650919570238307791960, 16.75471643768571878345340564262, 17.16454599957755393765447749956, 17.85321210474416707975090884465, 18.31885601954766306915009708095

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