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Properties

Label 1-3724-3724.1315-r0-0-0
Degree $1$
Conductor $3724$
Sign $-0.982 - 0.187i$
Analytic cond. $17.2941$
Root an. cond. $17.2941$
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  + (−0.318 + 0.947i)3-s + (0.661 + 0.749i)5-s + (−0.797 − 0.603i)9-s + (0.733 + 0.680i)11-s + (−0.542 + 0.840i)13-s + (−0.921 + 0.388i)15-s + (0.969 + 0.246i)17-s + (−0.270 + 0.962i)23-s + (−0.124 + 0.992i)25-s + (0.826 − 0.563i)27-s + (−0.411 + 0.911i)29-s + (−0.5 − 0.866i)31-s + (−0.878 + 0.478i)33-s + (−0.900 + 0.433i)37-s + (−0.623 − 0.781i)39-s + ⋯
L(s)  = 1  + (−0.318 + 0.947i)3-s + (0.661 + 0.749i)5-s + (−0.797 − 0.603i)9-s + (0.733 + 0.680i)11-s + (−0.542 + 0.840i)13-s + (−0.921 + 0.388i)15-s + (0.969 + 0.246i)17-s + (−0.270 + 0.962i)23-s + (−0.124 + 0.992i)25-s + (0.826 − 0.563i)27-s + (−0.411 + 0.911i)29-s + (−0.5 − 0.866i)31-s + (−0.878 + 0.478i)33-s + (−0.900 + 0.433i)37-s + (−0.623 − 0.781i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.982 - 0.187i$
Analytic conductor: \(17.2941\)
Root analytic conductor: \(17.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (0:\ ),\ -0.982 - 0.187i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1251784223 + 1.320775738i\)
\(L(\frac12)\) \(\approx\) \(-0.1251784223 + 1.320775738i\)
\(L(1)\) \(\approx\) \(0.8040923413 + 0.6227366180i\)
\(L(1)\) \(\approx\) \(0.8040923413 + 0.6227366180i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.318 + 0.947i)T \)
5 \( 1 + (0.661 + 0.749i)T \)
11 \( 1 + (0.733 + 0.680i)T \)
13 \( 1 + (-0.542 + 0.840i)T \)
17 \( 1 + (0.969 + 0.246i)T \)
23 \( 1 + (-0.270 + 0.962i)T \)
29 \( 1 + (-0.411 + 0.911i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (0.318 - 0.947i)T \)
43 \( 1 + (-0.878 + 0.478i)T \)
47 \( 1 + (0.456 + 0.889i)T \)
53 \( 1 + (0.270 - 0.962i)T \)
59 \( 1 + (-0.0249 + 0.999i)T \)
61 \( 1 + (0.583 + 0.811i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.583 - 0.811i)T \)
73 \( 1 + (-0.542 - 0.840i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.733 + 0.680i)T \)
89 \( 1 + (0.124 - 0.992i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−18.25428842279234543037167854003, −17.532880562854070230853822278028, −16.84962425190899321719763747529, −16.63501410223946993330313495451, −15.64659917791375089795563216517, −14.49055973399795852888551869622, −14.110895093196511005218949971908, −13.37128776321248941170974558810, −12.66437254623726579523559568264, −12.21051781733178181554344016824, −11.55391222505187949591578279918, −10.592683936234838633406646183, −9.93163607914863456846169271864, −9.02714289859543480393926805835, −8.37022280425240519401715439171, −7.756335633896422924490983031006, −6.83871599284589135782940168001, −6.14085714023302239360536101643, −5.46827248657531786744590463124, −4.972728079415221609416771562349, −3.75941661010745996829734783395, −2.805033899030820852323126526940, −1.975802629349698273087465117472, −1.143174776027320806120681795495, −0.40787820040796737881850061718, 1.415501847865278921038570567890, 2.1758634143271708127642851149, 3.25641533755427712772221180941, 3.80648017779494399984131753068, 4.68471836436932582832373188310, 5.50129737498663809165033046519, 6.07701390264498576907748649349, 6.95947102695465635353547672453, 7.512209096098744709821983126056, 8.82572227229972579307890047737, 9.3948893408858250007070836744, 9.96116537284956368401650976782, 10.47074426543445209190282776082, 11.41293596317352887105788466189, 11.83944356234856215997045086373, 12.67951804818217109129895821718, 13.76693989528989115981323005822, 14.3608203319921577418825295377, 14.85001697690647092358526911381, 15.40146489894584679754219564978, 16.50344363518304421057410279483, 16.82261684950155567935421384262, 17.58546753394421141362743549360, 18.07564098630821964484408273799, 19.07737302226107144118659636902

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