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Properties

Label 1-4235-4235.164-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  + (0.928 + 0.371i)2-s + (−0.5 − 0.866i)3-s + (0.723 + 0.690i)4-s + (−0.142 − 0.989i)6-s + (0.415 + 0.909i)8-s + (−0.5 + 0.866i)9-s + (0.235 − 0.971i)12-s + (0.959 − 0.281i)13-s + (0.0475 + 0.998i)16-s + (0.327 − 0.945i)17-s + (−0.786 + 0.618i)18-s + (−0.327 − 0.945i)19-s + (−0.0475 − 0.998i)23-s + (0.580 − 0.814i)24-s + (0.995 + 0.0950i)26-s + 27-s + ⋯
L(s)  = 1  + (0.928 + 0.371i)2-s + (−0.5 − 0.866i)3-s + (0.723 + 0.690i)4-s + (−0.142 − 0.989i)6-s + (0.415 + 0.909i)8-s + (−0.5 + 0.866i)9-s + (0.235 − 0.971i)12-s + (0.959 − 0.281i)13-s + (0.0475 + 0.998i)16-s + (0.327 − 0.945i)17-s + (−0.786 + 0.618i)18-s + (−0.327 − 0.945i)19-s + (−0.0475 − 0.998i)23-s + (0.580 − 0.814i)24-s + (0.995 + 0.0950i)26-s + 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: $\chi_{4235} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.245 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.910536832 - 1.487194213i\)
\(L(\frac12)\) \(\approx\) \(1.910536832 - 1.487194213i\)
\(L(1)\) \(\approx\) \(1.546639810 - 0.2253186831i\)
\(L(1)\) \(\approx\) \(1.546639810 - 0.2253186831i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.928 + 0.371i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.959 - 0.281i)T \)
17 \( 1 + (0.327 - 0.945i)T \)
19 \( 1 + (-0.327 - 0.945i)T \)
23 \( 1 + (-0.0475 - 0.998i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (-0.235 - 0.971i)T \)
37 \( 1 + (-0.723 + 0.690i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (-0.786 - 0.618i)T \)
53 \( 1 + (-0.0475 + 0.998i)T \)
59 \( 1 + (-0.928 + 0.371i)T \)
61 \( 1 + (-0.786 - 0.618i)T \)
67 \( 1 + (0.786 - 0.618i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (0.888 + 0.458i)T \)
79 \( 1 + (-0.580 - 0.814i)T \)
83 \( 1 + (-0.841 - 0.540i)T \)
89 \( 1 + (-0.981 + 0.189i)T \)
97 \( 1 + (0.415 + 0.909i)T \)
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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

−18.53905235734843623802712141733, −17.825745201029055662142178130391, −16.91616130182461377164909589192, −16.309109075360684141139427453609, −15.73160325731584319674049816582, −15.17268804714125787290727374132, −14.32053037220179164821914870438, −14.00492556759721516654030573548, −12.8765144438036410477715333118, −12.41879004668218104149237796326, −11.64704862737843616496348808973, −11.01080234251904720108817410892, −10.45697108804601614627922837622, −9.89785128127037809253913987324, −8.992332534532759782565213325787, −8.21534221914168208358253942730, −7.06748770748171948635055641845, −6.25985589201129816833469866802, −5.78487722800695812228705925778, −5.07175730397401641963024141548, −4.287964597995601827291132101445, −3.55117020605288363805024894577, −3.22943265829347566673307353476, −1.82494981873314937615928963870, −1.19006389553023723298537815003, 0.50854377600496125039986514734, 1.6017051533808126092202824394, 2.53776312676552857573580206446, 3.09583699017168256149572540138, 4.247441929991894206948832161901, 4.88858819998000181083494354629, 5.67230301380127544156918198075, 6.32456646055989994549532534119, 6.83891340508898365302024873219, 7.62910736425309735618525054313, 8.242729568284830462623550493312, 8.99088534468095325033102836470, 10.2756409313030981664934759970, 11.0391386257101649044605232160, 11.53284298211348503419583201760, 12.23995150820186649597385721683, 12.84459553228722154036016027955, 13.564894094114064810936254371103, 13.87570569141886880662134929510, 14.74086674373843615051153139423, 15.64732290216568379533392092990, 16.04149156707904424019110250501, 16.99390974140457485121080055401, 17.29562706920842930275075432261, 18.29195567335745506124498368136

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