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Properties

Label 1-4235-4235.2999-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.245 - 0.969i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.0665 + 0.997i)2-s + (−0.104 + 0.994i)3-s + (−0.991 − 0.132i)4-s + (−0.985 − 0.170i)6-s + (0.198 − 0.980i)8-s + (−0.978 − 0.207i)9-s + (0.235 − 0.971i)12-s + (0.564 + 0.825i)13-s + (0.964 + 0.263i)16-s + (−0.797 − 0.603i)17-s + (0.272 − 0.962i)18-s + (−0.290 + 0.956i)19-s + (−0.0475 − 0.998i)23-s + (0.953 + 0.299i)24-s + (−0.861 + 0.508i)26-s + (0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.0665 + 0.997i)2-s + (−0.104 + 0.994i)3-s + (−0.991 − 0.132i)4-s + (−0.985 − 0.170i)6-s + (0.198 − 0.980i)8-s + (−0.978 − 0.207i)9-s + (0.235 − 0.971i)12-s + (0.564 + 0.825i)13-s + (0.964 + 0.263i)16-s + (−0.797 − 0.603i)17-s + (0.272 − 0.962i)18-s + (−0.290 + 0.956i)19-s + (−0.0475 − 0.998i)23-s + (0.953 + 0.299i)24-s + (−0.861 + 0.508i)26-s + (0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.245 - 0.969i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.245 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2830247561 + 0.2203112614i\)
\(L(\frac12)\) \(\approx\) \(-0.2830247561 + 0.2203112614i\)
\(L(1)\) \(\approx\) \(0.4832614572 + 0.5682226606i\)
\(L(1)\) \(\approx\) \(0.4832614572 + 0.5682226606i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0665 + 0.997i)T \)
3 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.564 + 0.825i)T \)
17 \( 1 + (-0.797 - 0.603i)T \)
19 \( 1 + (-0.290 + 0.956i)T \)
23 \( 1 + (-0.0475 - 0.998i)T \)
29 \( 1 + (-0.0855 + 0.996i)T \)
31 \( 1 + (0.851 - 0.524i)T \)
37 \( 1 + (0.179 - 0.983i)T \)
41 \( 1 + (-0.466 + 0.884i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (0.272 + 0.962i)T \)
53 \( 1 + (-0.964 + 0.263i)T \)
59 \( 1 + (0.532 - 0.846i)T \)
61 \( 1 + (-0.830 + 0.556i)T \)
67 \( 1 + (0.786 - 0.618i)T \)
71 \( 1 + (0.516 + 0.856i)T \)
73 \( 1 + (-0.988 + 0.151i)T \)
79 \( 1 + (0.595 - 0.803i)T \)
83 \( 1 + (-0.774 + 0.633i)T \)
89 \( 1 + (-0.981 + 0.189i)T \)
97 \( 1 + (-0.736 + 0.676i)T \)
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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

−17.90408360592154393822485729803, −17.37241608266188980376658085892, −17.02509230663741481112794812179, −15.61794471939214054668133451414, −15.14408811176754299601163274501, −14.0202678863828571674989588963, −13.46526487988828939356754223829, −13.141182579465357193506947435767, −12.34071635723678699643149238387, −11.716329743387982988780214140569, −11.07992156285376903928529494150, −10.50732344413934926763433006121, −9.638334731944180207650990875993, −8.65857127393737162424057744437, −8.37558837308771835560454100948, −7.504539652121429856618955852652, −6.64243490491395050285595896239, −5.81985922198675827727778610047, −5.13261649094365564182046915811, −4.18342679197260789760107111752, −3.30852272322598299921367677687, −2.58598789089143840207342315198, −1.84906204846364636554029864447, −1.05074651198619724485057492281, −0.12031874519195723813169941710, 1.22066003704643958325706951716, 2.5476735307382743505684710168, 3.54077571279841016830172634285, 4.356555283864264872440631597149, 4.65778292542546872162040949475, 5.65982933613564226838988419839, 6.293937359476015662553023391000, 6.865118629024692189640058429803, 7.993566444609318480878208441742, 8.50414040675597867953736628403, 9.29441315275609942654363431468, 9.67005514875503209518879253057, 10.64554765363613773879039481630, 11.15111139209674209396509345844, 12.137086004806593953774934274444, 12.92599229282442798862952366374, 13.82505771268854484058732762005, 14.38444976898151764375630230486, 14.83248123135557825947953983863, 15.79734282120518585966218340420, 16.096922128420097173159064022611, 16.69223580801311452921863621316, 17.32843652091170443266697462111, 18.108383482551162968872470727184, 18.67959235917030347380904489512

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