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Properties

Label 1-4013-4013.97-r0-0-0
Degree $1$
Conductor $4013$
Sign $-0.933 - 0.358i$
Analytic cond. $18.6362$
Root an. cond. $18.6362$
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  + (0.753 + 0.657i)2-s + (0.279 + 0.960i)3-s + (0.135 + 0.990i)4-s + (0.465 − 0.885i)5-s + (−0.420 + 0.907i)6-s + (0.428 + 0.903i)7-s + (−0.548 + 0.835i)8-s + (−0.843 + 0.537i)9-s + (0.932 − 0.361i)10-s + (0.951 − 0.308i)11-s + (−0.913 + 0.407i)12-s + (0.947 + 0.320i)13-s + (−0.270 + 0.962i)14-s + (0.979 + 0.199i)15-s + (−0.963 + 0.269i)16-s + (−0.906 + 0.421i)17-s + ⋯
L(s)  = 1  + (0.753 + 0.657i)2-s + (0.279 + 0.960i)3-s + (0.135 + 0.990i)4-s + (0.465 − 0.885i)5-s + (−0.420 + 0.907i)6-s + (0.428 + 0.903i)7-s + (−0.548 + 0.835i)8-s + (−0.843 + 0.537i)9-s + (0.932 − 0.361i)10-s + (0.951 − 0.308i)11-s + (−0.913 + 0.407i)12-s + (0.947 + 0.320i)13-s + (−0.270 + 0.962i)14-s + (0.979 + 0.199i)15-s + (−0.963 + 0.269i)16-s + (−0.906 + 0.421i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4013\)
Sign: $-0.933 - 0.358i$
Analytic conductor: \(18.6362\)
Root analytic conductor: \(18.6362\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4013} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4013,\ (0:\ ),\ -0.933 - 0.358i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5082717144 + 2.744878385i\)
\(L(\frac12)\) \(\approx\) \(-0.5082717144 + 2.744878385i\)
\(L(1)\) \(\approx\) \(1.140058701 + 1.374259689i\)
\(L(1)\) \(\approx\) \(1.140058701 + 1.374259689i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4013 \( 1 \)
good2 \( 1 + (0.753 + 0.657i)T \)
3 \( 1 + (0.279 + 0.960i)T \)
5 \( 1 + (0.465 - 0.885i)T \)
7 \( 1 + (0.428 + 0.903i)T \)
11 \( 1 + (0.951 - 0.308i)T \)
13 \( 1 + (0.947 + 0.320i)T \)
17 \( 1 + (-0.906 + 0.421i)T \)
19 \( 1 + (-0.151 + 0.988i)T \)
23 \( 1 + (-0.810 - 0.586i)T \)
29 \( 1 + (0.980 + 0.196i)T \)
31 \( 1 + (-0.924 + 0.381i)T \)
37 \( 1 + (-0.851 - 0.523i)T \)
41 \( 1 + (-0.607 + 0.794i)T \)
43 \( 1 + (-0.582 - 0.812i)T \)
47 \( 1 + (-0.101 + 0.994i)T \)
53 \( 1 + (-0.982 - 0.183i)T \)
59 \( 1 + (-0.541 + 0.841i)T \)
61 \( 1 + (0.203 - 0.979i)T \)
67 \( 1 + (0.874 - 0.485i)T \)
71 \( 1 + (0.915 + 0.401i)T \)
73 \( 1 + (-0.519 + 0.854i)T \)
79 \( 1 + (0.431 - 0.902i)T \)
83 \( 1 + (-0.684 + 0.729i)T \)
89 \( 1 + (0.787 + 0.616i)T \)
97 \( 1 + (-0.212 + 0.977i)T \)
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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

−18.04790242718428957782032756137, −17.77775951488811857577883433447, −17.03120467665951595278658993320, −15.73551421463438697253692412011, −15.094944851539043726136808199318, −14.36510813079429195115576102117, −13.74614837906587556064666177837, −13.57563028640033353284803691497, −12.81103721025521909269509436804, −11.78672805786229684156818551182, −11.34646696470999037668643668857, −10.785258794443680077397314675936, −9.94802198889309715049206842461, −9.1856473605471895651271182424, −8.334215557886664148459544488189, −7.22375284498237366900756990497, −6.749645758921381856799338907985, −6.28675524448941990294594923203, −5.36643843078840247547538142635, −4.36112677281349843922523823522, −3.57110300199851972880726633362, −2.9895968924391471629364919871, −1.902605790925945931244958193345, −1.626219976992775851810745119462, −0.49563969344047943042439640559, 1.61582644464064113384631920308, 2.25269072428204932013886668596, 3.43668769066541273842383803715, 3.993956989950059865283347047223, 4.70898594501665039970544198258, 5.3234968751153220228033054119, 6.12979114788506224186453022448, 6.45306614872802978198267426751, 8.020360334757762556726674007162, 8.55861459504928706446961260604, 8.838791146854696091013360683079, 9.60028743083297005351817434750, 10.70491056048420793378119221389, 11.421981925996890941564461902663, 12.15934335166429864291675664454, 12.69903010695751832705917538881, 13.726687050590571043216851391936, 14.15881010419740424955399881105, 14.70057804517247141533595358218, 15.58810113726186001684111448678, 16.03568656031340917087942176381, 16.56281177742994156611280803866, 17.27020695299272272565084669798, 17.861783728975376922130843276883, 18.78056213059346638271202583142

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