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Properties

Label 1-709-709.41-r1-0-0
Degree $1$
Conductor $709$
Sign $0.481 - 0.876i$
Analytic cond. $76.1926$
Root an. cond. $76.1926$
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  + (−0.775 − 0.631i)2-s + (−0.271 − 0.962i)3-s + (0.202 + 0.979i)4-s + (0.992 + 0.123i)5-s + (−0.396 + 0.917i)6-s + (−0.560 − 0.828i)7-s + (0.461 − 0.887i)8-s + (−0.852 + 0.522i)9-s + (−0.691 − 0.722i)10-s + (0.671 − 0.740i)11-s + (0.887 − 0.461i)12-s + (−0.899 + 0.437i)13-s + (−0.0886 + 0.996i)14-s + (−0.150 − 0.988i)15-s + (−0.917 + 0.396i)16-s + (0.952 + 0.305i)17-s + ⋯
L(s)  = 1  + (−0.775 − 0.631i)2-s + (−0.271 − 0.962i)3-s + (0.202 + 0.979i)4-s + (0.992 + 0.123i)5-s + (−0.396 + 0.917i)6-s + (−0.560 − 0.828i)7-s + (0.461 − 0.887i)8-s + (−0.852 + 0.522i)9-s + (−0.691 − 0.722i)10-s + (0.671 − 0.740i)11-s + (0.887 − 0.461i)12-s + (−0.899 + 0.437i)13-s + (−0.0886 + 0.996i)14-s + (−0.150 − 0.988i)15-s + (−0.917 + 0.396i)16-s + (0.952 + 0.305i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.481 - 0.876i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ 0.481 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.239821703 - 0.7330502410i\)
\(L(\frac12)\) \(\approx\) \(1.239821703 - 0.7330502410i\)
\(L(1)\) \(\approx\) \(0.7222352605 - 0.4096896203i\)
\(L(1)\) \(\approx\) \(0.7222352605 - 0.4096896203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.775 - 0.631i)T \)
3 \( 1 + (-0.271 - 0.962i)T \)
5 \( 1 + (0.992 + 0.123i)T \)
7 \( 1 + (-0.560 - 0.828i)T \)
11 \( 1 + (0.671 - 0.740i)T \)
13 \( 1 + (-0.899 + 0.437i)T \)
17 \( 1 + (0.952 + 0.305i)T \)
19 \( 1 + (0.842 - 0.537i)T \)
23 \( 1 + (0.891 + 0.453i)T \)
29 \( 1 + (-0.792 + 0.610i)T \)
31 \( 1 + (0.624 + 0.781i)T \)
37 \( 1 + (0.828 - 0.560i)T \)
41 \( 1 + (-0.703 + 0.710i)T \)
43 \( 1 + (-0.895 + 0.445i)T \)
47 \( 1 + (0.998 - 0.0532i)T \)
53 \( 1 + (-0.364 + 0.931i)T \)
59 \( 1 + (-0.887 - 0.461i)T \)
61 \( 1 + (0.582 + 0.813i)T \)
67 \( 1 + (0.999 + 0.0354i)T \)
71 \( 1 + (-0.691 + 0.722i)T \)
73 \( 1 + (-0.106 - 0.994i)T \)
79 \( 1 + (0.330 + 0.943i)T \)
83 \( 1 + (0.786 + 0.617i)T \)
89 \( 1 + (0.728 - 0.684i)T \)
97 \( 1 + (0.429 + 0.903i)T \)
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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

−22.44694377344543685402084030332, −21.96413916364406788810023387366, −20.69060194418372309353717647156, −20.31873670408063969898977825759, −19.06526033334180995336518331463, −18.35325380876384099916031112896, −17.30985135827538137943302588850, −16.950000631561549786057800154118, −16.18205316912996195329072873226, −15.06756675458657201180593779000, −14.81881984360129330322546258513, −13.76010444138410741834129407536, −12.39836414427387985463204923005, −11.60144305445272998673412098439, −10.3109289470834213167467302880, −9.59295628087182359505002508522, −9.515333723425080364562555253686, −8.37245227066982864900816984585, −7.11506859277330192785278773870, −6.11094615169950593436812725567, −5.46817000260481488335593545335, −4.76132764442911546948189525538, −3.10728671461384200898434597287, −2.04223366592557121022410055178, −0.59922038427476663710376314403, 0.87658601230357295982320181228, 1.40736614037698090439068100277, 2.68754416669386693738825614406, 3.4351820573224076220190283000, 5.07341563155433772874804812393, 6.29799907519156575646014928607, 7.00384652619361158777218612552, 7.70096230928056264556031197487, 8.97842330240249619476934916903, 9.61816890351723917541740879007, 10.550634001788818542453649344457, 11.36533558769574061181848432914, 12.23418251536004534681856407748, 13.098288568086814832143082590233, 13.70743343199341502213574105557, 14.50541279245242607104732508763, 16.32575591807231043397487496729, 16.957748778013418155222251149072, 17.29761329047196803916103918911, 18.31087064753842273499844444165, 19.00373074739607368844237246895, 19.669435493753063002527875191631, 20.35333807399338965166053012190, 21.58263884457364692831070763586, 22.02275966261588070948163089222

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