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Properties

Label 1-4235-4235.387-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.977 - 0.212i$
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  + (−0.0380 − 0.999i)2-s + (0.743 − 0.669i)3-s + (−0.997 + 0.0760i)4-s + (−0.696 − 0.717i)6-s + (0.113 + 0.993i)8-s + (0.104 − 0.994i)9-s + (−0.690 + 0.723i)12-s + (−0.336 + 0.941i)13-s + (0.988 − 0.151i)16-s + (0.730 + 0.683i)17-s + (−0.997 − 0.0665i)18-s + (0.483 − 0.875i)19-s + (−0.458 + 0.888i)23-s + (0.749 + 0.662i)24-s + (0.953 + 0.299i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.0380 − 0.999i)2-s + (0.743 − 0.669i)3-s + (−0.997 + 0.0760i)4-s + (−0.696 − 0.717i)6-s + (0.113 + 0.993i)8-s + (0.104 − 0.994i)9-s + (−0.690 + 0.723i)12-s + (−0.336 + 0.941i)13-s + (0.988 − 0.151i)16-s + (0.730 + 0.683i)17-s + (−0.997 − 0.0665i)18-s + (0.483 − 0.875i)19-s + (−0.458 + 0.888i)23-s + (0.749 + 0.662i)24-s + (0.953 + 0.299i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.977 - 0.212i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.977 - 0.212i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1706846642 - 1.585253451i\)
\(L(\frac12)\) \(\approx\) \(0.1706846642 - 1.585253451i\)
\(L(1)\) \(\approx\) \(0.8310209377 - 0.7742339736i\)
\(L(1)\) \(\approx\) \(0.8310209377 - 0.7742339736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0380 - 0.999i)T \)
3 \( 1 + (0.743 - 0.669i)T \)
13 \( 1 + (-0.336 + 0.941i)T \)
17 \( 1 + (0.730 + 0.683i)T \)
19 \( 1 + (0.483 - 0.875i)T \)
23 \( 1 + (-0.458 + 0.888i)T \)
29 \( 1 + (-0.921 - 0.389i)T \)
31 \( 1 + (-0.991 + 0.132i)T \)
37 \( 1 + (-0.524 - 0.851i)T \)
41 \( 1 + (0.985 + 0.170i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (0.997 - 0.0665i)T \)
53 \( 1 + (0.151 - 0.988i)T \)
59 \( 1 + (-0.345 + 0.938i)T \)
61 \( 1 + (0.532 - 0.846i)T \)
67 \( 1 + (-0.371 + 0.928i)T \)
71 \( 1 + (0.0855 - 0.996i)T \)
73 \( 1 + (-0.353 - 0.935i)T \)
79 \( 1 + (0.861 + 0.508i)T \)
83 \( 1 + (-0.0570 - 0.998i)T \)
89 \( 1 + (0.327 - 0.945i)T \)
97 \( 1 + (0.980 + 0.198i)T \)
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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

−18.57741789350274044363091449388, −18.07557140812622316363485928700, −16.98486014541890461928265478665, −16.60649743514428525829571500201, −15.922325038203498086336533046950, −15.32345535616323563021862421976, −14.62838025338549972269942213211, −14.23677553536815172352974454012, −13.54002663322701086892331819346, −12.75105434833903029057307486789, −12.09842884104639157716778727628, −10.832038144997621615078903936682, −10.17913355585466396174092595851, −9.63827945557881715381618241848, −8.951721861020037123489898091599, −8.20149233039936229094735044582, −7.6426753907197932776350703646, −7.09844652023790863853003071776, −5.87705233675884847311074872903, −5.37329567931738308795746976699, −4.6865715532316941546665182470, −3.75233402786819419377260266665, −3.265284361311898277193693911767, −2.222671952844647852172648516748, −0.97947219267257657944938045621, 0.451458299159221101080344524596, 1.63506199084938709647442863654, 1.94752265427080782803311354759, 2.9364228322075477396631092793, 3.632010464407756470123155595881, 4.230081724182806403842096978269, 5.301829195773750927684261992497, 6.05518462681091288164227827993, 7.206631097658777774911163577417, 7.64372406398925027996398097713, 8.52298178668329680538683142883, 9.27950425808688549450936389995, 9.558490235859157517510423194323, 10.52807129573424463161302622818, 11.41426444277958965263856331790, 11.91580340077048898596239385587, 12.61741404611876730158542788359, 13.25612425203523846399837326852, 13.81241914894948587674013468866, 14.50955913483010748616448330948, 14.97619755284180206126703049650, 16.06594847272983980906637043930, 16.96561716004743520399666065386, 17.6572382919019628597944525664, 18.23603462003968710967205966703

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