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Properties

Label 1-4235-4235.2127-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.238 - 0.971i$
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  + (0.999 − 0.0285i)2-s + (0.587 − 0.809i)3-s + (0.998 − 0.0570i)4-s + (0.564 − 0.825i)6-s + (0.996 − 0.0855i)8-s + (−0.309 − 0.951i)9-s + (0.540 − 0.841i)12-s + (−0.967 − 0.254i)13-s + (0.993 − 0.113i)16-s + (−0.170 + 0.985i)17-s + (−0.336 − 0.941i)18-s + (0.696 − 0.717i)19-s + (0.909 − 0.415i)23-s + (0.516 − 0.856i)24-s + (−0.974 − 0.226i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (0.999 − 0.0285i)2-s + (0.587 − 0.809i)3-s + (0.998 − 0.0570i)4-s + (0.564 − 0.825i)6-s + (0.996 − 0.0855i)8-s + (−0.309 − 0.951i)9-s + (0.540 − 0.841i)12-s + (−0.967 − 0.254i)13-s + (0.993 − 0.113i)16-s + (−0.170 + 0.985i)17-s + (−0.336 − 0.941i)18-s + (0.696 − 0.717i)19-s + (0.909 − 0.415i)23-s + (0.516 − 0.856i)24-s + (−0.974 − 0.226i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.238 - 0.971i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.238 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.569779293 - 3.277548727i\)
\(L(\frac12)\) \(\approx\) \(2.569779293 - 3.277548727i\)
\(L(1)\) \(\approx\) \(2.108940514 - 0.9553373778i\)
\(L(1)\) \(\approx\) \(2.108940514 - 0.9553373778i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.999 - 0.0285i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (-0.967 - 0.254i)T \)
17 \( 1 + (-0.170 + 0.985i)T \)
19 \( 1 + (0.696 - 0.717i)T \)
23 \( 1 + (0.909 - 0.415i)T \)
29 \( 1 + (0.466 + 0.884i)T \)
31 \( 1 + (-0.774 - 0.633i)T \)
37 \( 1 + (-0.931 - 0.362i)T \)
41 \( 1 + (-0.610 - 0.791i)T \)
43 \( 1 + (-0.755 + 0.654i)T \)
47 \( 1 + (0.336 - 0.941i)T \)
53 \( 1 + (0.113 - 0.993i)T \)
59 \( 1 + (0.610 - 0.791i)T \)
61 \( 1 + (0.0285 - 0.999i)T \)
67 \( 1 + (0.281 - 0.959i)T \)
71 \( 1 + (0.897 + 0.441i)T \)
73 \( 1 + (0.491 - 0.870i)T \)
79 \( 1 + (0.921 + 0.389i)T \)
83 \( 1 + (0.676 - 0.736i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (-0.856 - 0.516i)T \)
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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

−18.86604925554774568250228837638, −17.755708543972229515555988752485, −16.76654149580716591395831371918, −16.49093787674051616987153104722, −15.58138355086713660573476436262, −15.20122096953606804287043597123, −14.452074352205591383648084476543, −13.87980216452467653418043939202, −13.4373816704780524173190670125, −12.424495638477816769892152565072, −11.80251948210443727693701714994, −11.15952491224236598115750962306, −10.30392627007086310215627503749, −9.757892915434196509343299202149, −8.99272869716640844452479978266, −8.0410428631317742624045590312, −7.34076058790552710169209149333, −6.75016829396827373486872435857, −5.52096725219436099133424374314, −5.12532078267134334475276217368, −4.42838317156151354002784953588, −3.6380231458195797715730121498, −2.93025799896174846780844590643, −2.3588195792479339758199661887, −1.33114110094392147781851524928, 0.670065017360024493498911142823, 1.81914449093883007016683608553, 2.29165828571044738225698592780, 3.27460314593127947295067417886, 3.66488714637841709905642488717, 4.917329100201120318222704783510, 5.32539532056761827410952728139, 6.44974539149538929099934234633, 6.88988692528241161241540035762, 7.54081706385691863777957165117, 8.29730698025336981109513962002, 9.10394576853134582158942053896, 10.008558446067875249576183050968, 10.83799280064204451231478992502, 11.5663778308958877062873253786, 12.366221720829471775685820532376, 12.73851669906396058196925371385, 13.39676847573051914679441049810, 14.060988441938568759317645205233, 14.7602466052620039108932001620, 15.12133291502484857496024351936, 15.91083858173889214758042151490, 16.91251888445804365882272792113, 17.376190886730139017763056675597, 18.29320918806915299173497235413

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