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Properties

Label 1-3100-3100.203-r1-0-0
Degree $1$
Conductor $3100$
Sign $-0.367 + 0.930i$
Analytic cond. $333.141$
Root an. cond. $333.141$
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  + (0.406 + 0.913i)3-s + (0.207 − 0.978i)7-s + (−0.669 + 0.743i)9-s + (−0.978 + 0.207i)11-s + (0.207 + 0.978i)13-s + (0.406 + 0.913i)17-s + (−0.104 − 0.994i)19-s + (0.978 − 0.207i)21-s + (−0.587 − 0.809i)23-s + (−0.951 − 0.309i)27-s + (0.309 − 0.951i)29-s + (−0.587 − 0.809i)33-s + (0.994 − 0.104i)37-s + (−0.809 + 0.587i)39-s + (−0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)3-s + (0.207 − 0.978i)7-s + (−0.669 + 0.743i)9-s + (−0.978 + 0.207i)11-s + (0.207 + 0.978i)13-s + (0.406 + 0.913i)17-s + (−0.104 − 0.994i)19-s + (0.978 − 0.207i)21-s + (−0.587 − 0.809i)23-s + (−0.951 − 0.309i)27-s + (0.309 − 0.951i)29-s + (−0.587 − 0.809i)33-s + (0.994 − 0.104i)37-s + (−0.809 + 0.587i)39-s + (−0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3100\)    =    \(2^{2} \cdot 5^{2} \cdot 31\)
Sign: $-0.367 + 0.930i$
Analytic conductor: \(333.141\)
Root analytic conductor: \(333.141\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3100} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3100,\ (1:\ ),\ -0.367 + 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.012694261 + 1.488592384i\)
\(L(\frac12)\) \(\approx\) \(1.012694261 + 1.488592384i\)
\(L(1)\) \(\approx\) \(1.059419797 + 0.3204577158i\)
\(L(1)\) \(\approx\) \(1.059419797 + 0.3204577158i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.406 + 0.913i)T \)
7 \( 1 + (0.207 - 0.978i)T \)
11 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (0.207 + 0.978i)T \)
17 \( 1 + (0.406 + 0.913i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.587 - 0.809i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.994 - 0.104i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.406 + 0.913i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.207 - 0.978i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.207 + 0.978i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.406 + 0.913i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.587 + 0.809i)T \)
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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

−18.666399920698213901030031451365, −17.98325010978609408909390578001, −17.62419447340230847113518111779, −16.398518672151066203364538012667, −15.77448618667120483801418176738, −15.05539444991011141797650762955, −14.36105886418581885411161202474, −13.68642970760557511967233453082, −12.87751078909437301268717818736, −12.42930349499334150764428702629, −11.708014527193595443660316618224, −10.96158084029417608533434581201, −9.96033084576291873043724251124, −9.27233651240473441609052918359, −8.234775732321519139339317027314, −8.04719137646362742004027649613, −7.25108223480582658214005581336, −6.182821283295276727341386806590, −5.61336797890748235801456041275, −5.001147622707233423772345267371, −3.5258690704790054243648994275, −2.93431231799829396710419000982, −2.21663393273848338998902827603, −1.31872479211302660216643702787, −0.33013131446090417997883097141, 0.7479996644799269439939855563, 2.04183605265836153763410488050, 2.70522770736106258867382113530, 3.83349946504852400135005158014, 4.28027917599997081653537076009, 4.956286082599169389652971796125, 5.9242154274719913542125882174, 6.8250419000230119633167882470, 7.75603145597277699295449126647, 8.27672569543653159129600711440, 9.114351369603225993354525791007, 9.94302136336883159662982655592, 10.46591660703516683266130490905, 11.04612634619326458486516460272, 11.82532912604640665698849825107, 12.952481183083060339391591200081, 13.53519344701669977854580689325, 14.22592022118565643448826707731, 14.81509437465792407239982887270, 15.63213167542754089402045370243, 16.18151131923394195053685562866, 16.922795609447182174786520496583, 17.41628480887619234908279051452, 18.39519412029116730099261553308, 19.18651893002064161233704262437

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