Properties

Label 1-4235-4235.1153-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.0951 + 0.995i$
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  + (−0.983 + 0.179i)2-s + (0.743 − 0.669i)3-s + (0.935 − 0.353i)4-s + (−0.610 + 0.791i)6-s + (−0.856 + 0.516i)8-s + (0.104 − 0.994i)9-s + (0.458 − 0.888i)12-s + (−0.0570 + 0.998i)13-s + (0.749 − 0.662i)16-s + (−0.0380 + 0.999i)17-s + (0.0760 + 0.997i)18-s + (0.345 + 0.938i)19-s + (−0.0950 − 0.995i)23-s + (−0.290 + 0.956i)24-s + (−0.123 − 0.992i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.983 + 0.179i)2-s + (0.743 − 0.669i)3-s + (0.935 − 0.353i)4-s + (−0.610 + 0.791i)6-s + (−0.856 + 0.516i)8-s + (0.104 − 0.994i)9-s + (0.458 − 0.888i)12-s + (−0.0570 + 0.998i)13-s + (0.749 − 0.662i)16-s + (−0.0380 + 0.999i)17-s + (0.0760 + 0.997i)18-s + (0.345 + 0.938i)19-s + (−0.0950 − 0.995i)23-s + (−0.290 + 0.956i)24-s + (−0.123 − 0.992i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0951 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0951 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.0951 + 0.995i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.0951 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6161361724 + 0.5600604116i\)
\(L(\frac12)\) \(\approx\) \(0.6161361724 + 0.5600604116i\)
\(L(1)\) \(\approx\) \(0.7965928152 + 0.01319557620i\)
\(L(1)\) \(\approx\) \(0.7965928152 + 0.01319557620i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.983 - 0.179i)T \)
3 \( 1 + (-0.743 + 0.669i)T \)
13 \( 1 + (0.0570 - 0.998i)T \)
17 \( 1 + (0.0380 - 0.999i)T \)
19 \( 1 + (-0.345 - 0.938i)T \)
23 \( 1 + (0.0950 + 0.995i)T \)
29 \( 1 + (0.897 - 0.441i)T \)
31 \( 1 + (0.988 + 0.151i)T \)
37 \( 1 + (0.263 + 0.964i)T \)
41 \( 1 + (-0.0285 + 0.999i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 + (0.0760 - 0.997i)T \)
53 \( 1 + (0.662 - 0.749i)T \)
59 \( 1 + (-0.879 - 0.475i)T \)
61 \( 1 + (-0.761 - 0.647i)T \)
67 \( 1 + (0.971 + 0.235i)T \)
71 \( 1 + (-0.696 - 0.717i)T \)
73 \( 1 + (-0.803 - 0.595i)T \)
79 \( 1 + (0.820 - 0.572i)T \)
83 \( 1 + (0.491 - 0.870i)T \)
89 \( 1 + (-0.928 + 0.371i)T \)
97 \( 1 + (-0.226 - 0.974i)T \)
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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

−18.229332078520872117354523688088, −17.63018060042445523239060348531, −16.82771047515043236353068782458, −16.190024862225554309370441170036, −15.54232137666824589492554989647, −15.1189574258666838012297373589, −14.33911678908532104367776655813, −13.27956345110949042278009805462, −12.9806274748510361755530048777, −11.58011673972026254065432677229, −11.39671068069647919497233410439, −10.418104579076200908190824458670, −9.822830407708642648030516141190, −9.32789312010593876637579513916, −8.66154154576335371342566854821, −7.83757564321872499358713420869, −7.45426980493193604727086622463, −6.55295406613453404927510573456, −5.42200664229470137060190648582, −4.8460018080668767574375738560, −3.529160366868975966844639719507, −3.19610797967770239854417829263, −2.343210339204050428260365131759, −1.52409270381518488572217991139, −0.287384781088091313122204079522, 1.07537833911854892960449045545, 1.875125463308822963958689114616, 2.30608470067069064860584511302, 3.458968518074685027567312771700, 4.07142352942805540516978162205, 5.51053735390966240098281896027, 6.16200697291344125225612309007, 6.95073126016690765721478287690, 7.433187136839611305036348029252, 8.222371458077712553384583192503, 8.818298049490085482314250045368, 9.33293118902540679970959014881, 10.15068255008032983424505629652, 10.86862757464291443140093842540, 11.69385852964082959694362807763, 12.42113873634786745880993547286, 12.91925838307884463725725744252, 14.14584842004503054760758018092, 14.41440700257915083224421443225, 15.109466006166757982410938963567, 15.944239833873819558173119154936, 16.64781562665536995457822150983, 17.20340576475723733271292046569, 18.0533535962181783498245060490, 18.63210481357252396608956619214

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