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Properties

Label 1-66e2-4356.119-r0-0-0
Degree $1$
Conductor $4356$
Sign $-0.949 + 0.312i$
Analytic cond. $20.2291$
Root an. cond. $20.2291$
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  + (−0.999 − 0.0380i)5-s + (−0.123 − 0.992i)7-s + (0.00951 − 0.999i)13-s + (−0.941 + 0.336i)17-s + (0.0285 − 0.999i)19-s + (0.981 − 0.189i)23-s + (0.997 + 0.0760i)25-s + (0.432 − 0.901i)29-s + (−0.749 − 0.662i)31-s + (0.0855 + 0.996i)35-s + (−0.736 + 0.676i)37-s + (−0.964 − 0.263i)41-s + (0.786 − 0.618i)43-s + (−0.935 − 0.353i)47-s + (−0.969 + 0.244i)49-s + ⋯
L(s)  = 1  + (−0.999 − 0.0380i)5-s + (−0.123 − 0.992i)7-s + (0.00951 − 0.999i)13-s + (−0.941 + 0.336i)17-s + (0.0285 − 0.999i)19-s + (0.981 − 0.189i)23-s + (0.997 + 0.0760i)25-s + (0.432 − 0.901i)29-s + (−0.749 − 0.662i)31-s + (0.0855 + 0.996i)35-s + (−0.736 + 0.676i)37-s + (−0.964 − 0.263i)41-s + (0.786 − 0.618i)43-s + (−0.935 − 0.353i)47-s + (−0.969 + 0.244i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4356\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2}\)
Sign: $-0.949 + 0.312i$
Analytic conductor: \(20.2291\)
Root analytic conductor: \(20.2291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4356} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4356,\ (0:\ ),\ -0.949 + 0.312i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08884735450 - 0.5537207950i\)
\(L(\frac12)\) \(\approx\) \(-0.08884735450 - 0.5537207950i\)
\(L(1)\) \(\approx\) \(0.7146816841 - 0.2590975314i\)
\(L(1)\) \(\approx\) \(0.7146816841 - 0.2590975314i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (-0.999 - 0.0380i)T \)
7 \( 1 + (-0.123 - 0.992i)T \)
13 \( 1 + (0.00951 - 0.999i)T \)
17 \( 1 + (-0.941 + 0.336i)T \)
19 \( 1 + (0.0285 - 0.999i)T \)
23 \( 1 + (0.981 - 0.189i)T \)
29 \( 1 + (0.432 - 0.901i)T \)
31 \( 1 + (-0.749 - 0.662i)T \)
37 \( 1 + (-0.736 + 0.676i)T \)
41 \( 1 + (-0.964 - 0.263i)T \)
43 \( 1 + (0.786 - 0.618i)T \)
47 \( 1 + (-0.935 - 0.353i)T \)
53 \( 1 + (-0.974 + 0.226i)T \)
59 \( 1 + (-0.710 - 0.703i)T \)
61 \( 1 + (0.548 + 0.836i)T \)
67 \( 1 + (-0.0475 - 0.998i)T \)
71 \( 1 + (0.610 - 0.791i)T \)
73 \( 1 + (0.516 - 0.856i)T \)
79 \( 1 + (0.969 + 0.244i)T \)
83 \( 1 + (-0.905 + 0.424i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (0.999 - 0.0380i)T \)
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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

−18.68719601167015077156937771474, −18.24309057867484657740312051951, −17.34156993044981832223378180661, −16.447649023627120420028022958618, −15.93600499833620317478194876683, −15.49166483543177027142429242704, −14.51402907381790901811393172304, −14.30158822710546389042441189035, −13.02323156196615653683654724364, −12.55983433355519286155683351255, −11.82812484251387716653156881236, −11.314665009667854404140475447369, −10.68080085497747436260645171999, −9.61376108773612786180603823769, −8.88567255457822688145721428722, −8.53945745807741614772245275694, −7.57074776617024072859194363596, −6.86251953106693509945089624082, −6.277843863614295657941923353230, −5.17036639436006706561618162169, −4.702886388229611596460159601373, −3.70397552414589960126786162212, −3.12137365433248611696691581201, −2.16181653262116386518991304548, −1.31757252071389657359555789709, 0.197784721229819806051694633520, 0.85395614059092676696359629201, 2.09407486755900812621032530745, 3.17972638058495149461390669581, 3.622919295589384091943432534055, 4.61235291848331186063483574956, 4.962482045253546479011705061661, 6.23993963083623622438149176707, 6.909988406877178812214017943376, 7.520894210163413519221190414482, 8.19052122835774020123147156083, 8.87303936494141166181270440129, 9.73959611658292750487678270999, 10.72231753006865902559794685293, 10.92534950078538398683337205640, 11.7402101959388957011040933203, 12.61610440774042205832066672348, 13.19039629361150199830320752769, 13.73955579348266363538681655075, 14.73103621493302084216563544673, 15.494592309314592892534066423997, 15.60901790286483904797113193044, 16.76033076195746555485906179957, 17.1493935429963071757691345655, 17.87257383923406216801354515722

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