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Properties

Label 1-712-712.589-r0-0-0
Degree $1$
Conductor $712$
Sign $0.484 - 0.874i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
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  + i·3-s + 5-s − i·7-s − 9-s − 11-s − i·13-s + i·15-s − 17-s − i·19-s + 21-s − i·23-s + 25-s − i·27-s − i·29-s + i·31-s + ⋯
L(s)  = 1  + i·3-s + 5-s − i·7-s − 9-s − 11-s − i·13-s + i·15-s − 17-s − i·19-s + 21-s − i·23-s + 25-s − i·27-s − i·29-s + i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.484 - 0.874i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.484 - 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9923229046 - 0.5845542578i\)
\(L(\frac12)\) \(\approx\) \(0.9923229046 - 0.5845542578i\)
\(L(1)\) \(\approx\) \(1.030806508 + 0.02006290974i\)
\(L(1)\) \(\approx\) \(1.030806508 + 0.02006290974i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + iT \)
11 \( 1 \)
13 \( 1 + T \)
17 \( 1 \)
19 \( 1 - iT \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 - T \)
41 \( 1 \)
43 \( 1 - iT \)
47 \( 1 \)
53 \( 1 + iT \)
59 \( 1 \)
61 \( 1 - T \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 \)
79 \( 1 + T \)
83 \( 1 \)
97 \( 1 \)
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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

−22.7134637690119825477890707832, −21.87192967031022317519360424487, −21.20266569619869504360113526715, −20.3326061778071568004871845172, −19.26408477137964806057875093668, −18.37308638096531660284638625113, −18.217335500080612512541986932, −17.18022526630679418394957857537, −16.3481504552660690752927841238, −15.21109741832604668551166915326, −14.360820710602582193265247711749, −13.43622111089149007876723276953, −13.0020514120542095468259373853, −11.99423611079465178846276366752, −11.26534258442784542597742049634, −10.08632018092950224615570755783, −9.10626631224861697753976079053, −8.43758002055834600351814938001, −7.35492290210341989860601769382, −6.36188226224367552714621050692, −5.750086386357635365920252174001, −4.87037519900225915404307571737, −3.11577813634452948341335212667, −2.16532475499078018718019833264, −1.60926292325836280763583486639, 0.50928086928146594961953831011, 2.30751204684047985477464852871, 3.09909070830392569954641910464, 4.389603526543673357685277943629, 5.06384852380801544395191025624, 5.994616238480582606070128617743, 7.04219672184736408896358149557, 8.23634670189424836276539597687, 9.093722874772176593203510652191, 10.12079736100870050116832954334, 10.52612511730735817236375081994, 11.20515496411674912402823200378, 12.797816812380366771886757213289, 13.4154797348185026332770668146, 14.20254067567123174357475235692, 15.13900706061475738714620387775, 15.9280438653054942291538728025, 16.70991959705333522164343484849, 17.73914868680344337520606691237, 17.852975073383508263069204102035, 19.509862239383157820215991734187, 20.24776466375477880837744346659, 20.90239090437281616424168144366, 21.50845676737594517940864802888, 22.4976647924325748605097262768

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