Properties

Label 1-4235-4235.2119-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.0971 - 0.995i$
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  + (0.964 − 0.263i)2-s + (0.913 − 0.406i)3-s + (0.861 − 0.508i)4-s + (0.774 − 0.633i)6-s + (0.696 − 0.717i)8-s + (0.669 − 0.743i)9-s + (0.580 − 0.814i)12-s + (0.736 − 0.676i)13-s + (0.483 − 0.875i)16-s + (0.851 + 0.524i)17-s + (0.449 − 0.893i)18-s + (0.380 − 0.924i)19-s + (−0.981 − 0.189i)23-s + (0.345 − 0.938i)24-s + (0.532 − 0.846i)26-s + (0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (0.964 − 0.263i)2-s + (0.913 − 0.406i)3-s + (0.861 − 0.508i)4-s + (0.774 − 0.633i)6-s + (0.696 − 0.717i)8-s + (0.669 − 0.743i)9-s + (0.580 − 0.814i)12-s + (0.736 − 0.676i)13-s + (0.483 − 0.875i)16-s + (0.851 + 0.524i)17-s + (0.449 − 0.893i)18-s + (0.380 − 0.924i)19-s + (−0.981 − 0.189i)23-s + (0.345 − 0.938i)24-s + (0.532 − 0.846i)26-s + (0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0971 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0971 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.0971 - 0.995i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.0971 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.658226278 - 4.032785926i\)
\(L(\frac12)\) \(\approx\) \(3.658226278 - 4.032785926i\)
\(L(1)\) \(\approx\) \(2.476977539 - 1.243598066i\)
\(L(1)\) \(\approx\) \(2.476977539 - 1.243598066i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.964 + 0.263i)T \)
3 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (-0.736 + 0.676i)T \)
17 \( 1 + (-0.851 - 0.524i)T \)
19 \( 1 + (-0.380 + 0.924i)T \)
23 \( 1 + (0.981 + 0.189i)T \)
29 \( 1 + (0.941 - 0.336i)T \)
31 \( 1 + (-0.595 + 0.803i)T \)
37 \( 1 + (0.749 - 0.662i)T \)
41 \( 1 + (0.362 + 0.931i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (-0.449 - 0.893i)T \)
53 \( 1 + (0.483 + 0.875i)T \)
59 \( 1 + (-0.625 - 0.780i)T \)
61 \( 1 + (0.710 - 0.703i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (0.564 - 0.825i)T \)
73 \( 1 + (0.820 + 0.572i)T \)
79 \( 1 + (-0.830 - 0.556i)T \)
83 \( 1 + (-0.921 + 0.389i)T \)
89 \( 1 + (0.723 + 0.690i)T \)
97 \( 1 + (0.985 - 0.170i)T \)
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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

−18.762435251947919235137747803, −17.89653472755344018985884415573, −16.72532857200429814119685503217, −16.383902354243870478518560062296, −15.70052780299770864367523397421, −15.15102952779720248979661028553, −14.23100499789636100719669641351, −14.00072388317155235955734696763, −13.433985280227602774633276146160, −12.47176384823410424894649998475, −11.93786168614017263536771705935, −11.109605856569219681775963964412, −10.2971320983332097134778004517, −9.64968709173745130817095184805, −8.72084024960518109328251465588, −8.046253820665402938677353158846, −7.44944180914598282129830647520, −6.672607733988028962889399266, −5.75599986351987093374903333361, −5.131598292549996095791107091027, −4.193994402766878263808939279761, −3.64900388778856770674870637017, −3.10088337039728229106852728851, −2.053853140439136472914769522833, −1.490188053207430903873390593433, 0.89442637538487742567647058470, 1.64906634777758995703005643994, 2.49219324701959125949395862693, 3.24620966188329793028480816747, 3.75995352118147407341821538939, 4.580349621710014791991368098898, 5.579210362190054369456859099726, 6.1772186548870474496777686679, 6.99097698268734970725270389532, 7.73379349388134651423690456525, 8.31229690525382042210141362383, 9.28967868039104325812712360327, 10.03529507189863681056538656028, 10.67310820148819956612245981499, 11.59216263355926097487443164216, 12.209365780905828280696593700495, 12.97437592069812110910567231816, 13.37968069888062013204592721307, 14.06364618768760232437212644798, 14.65968510094744623998267438840, 15.364606442189110882904179259733, 15.791187942022208715066290659457, 16.65631719463854467218709832495, 17.65888873607354334845229400910, 18.392217407038386995497361481140

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