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Properties

Label 1-709-709.63-r0-0-0
Degree $1$
Conductor $709$
Sign $0.398 + 0.917i$
Analytic cond. $3.29258$
Root an. cond. $3.29258$
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  + (−0.931 − 0.364i)2-s + (0.288 + 0.957i)3-s + (0.734 + 0.678i)4-s + (0.802 + 0.596i)5-s + (0.0797 − 0.996i)6-s + (0.861 + 0.507i)7-s + (−0.437 − 0.899i)8-s + (−0.833 + 0.552i)9-s + (−0.530 − 0.847i)10-s + (−0.0266 − 0.999i)11-s + (−0.437 + 0.899i)12-s + (0.861 − 0.507i)13-s + (−0.617 − 0.786i)14-s + (−0.339 + 0.940i)15-s + (0.0797 + 0.996i)16-s + (−0.0266 + 0.999i)17-s + ⋯
L(s)  = 1  + (−0.931 − 0.364i)2-s + (0.288 + 0.957i)3-s + (0.734 + 0.678i)4-s + (0.802 + 0.596i)5-s + (0.0797 − 0.996i)6-s + (0.861 + 0.507i)7-s + (−0.437 − 0.899i)8-s + (−0.833 + 0.552i)9-s + (−0.530 − 0.847i)10-s + (−0.0266 − 0.999i)11-s + (−0.437 + 0.899i)12-s + (0.861 − 0.507i)13-s + (−0.617 − 0.786i)14-s + (−0.339 + 0.940i)15-s + (0.0797 + 0.996i)16-s + (−0.0266 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.398 + 0.917i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.398 + 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.144935588 + 0.7505358546i\)
\(L(\frac12)\) \(\approx\) \(1.144935588 + 0.7505358546i\)
\(L(1)\) \(\approx\) \(0.9680297188 + 0.3157977352i\)
\(L(1)\) \(\approx\) \(0.9680297188 + 0.3157977352i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.931 - 0.364i)T \)
3 \( 1 + (0.288 + 0.957i)T \)
5 \( 1 + (0.802 + 0.596i)T \)
7 \( 1 + (0.861 + 0.507i)T \)
11 \( 1 + (-0.0266 - 0.999i)T \)
13 \( 1 + (0.861 - 0.507i)T \)
17 \( 1 + (-0.0266 + 0.999i)T \)
19 \( 1 + (-0.437 - 0.899i)T \)
23 \( 1 + (0.388 - 0.921i)T \)
29 \( 1 + (0.977 + 0.211i)T \)
31 \( 1 + (-0.339 + 0.940i)T \)
37 \( 1 + (0.861 + 0.507i)T \)
41 \( 1 + (0.910 + 0.413i)T \)
43 \( 1 + (0.949 + 0.314i)T \)
47 \( 1 + (0.388 - 0.921i)T \)
53 \( 1 + (-0.339 + 0.940i)T \)
59 \( 1 + (-0.437 - 0.899i)T \)
61 \( 1 + (-0.998 + 0.0532i)T \)
67 \( 1 + (-0.964 + 0.263i)T \)
71 \( 1 + (-0.530 + 0.847i)T \)
73 \( 1 + (-0.697 - 0.716i)T \)
79 \( 1 + (0.574 + 0.818i)T \)
83 \( 1 + (0.484 - 0.874i)T \)
89 \( 1 + (-0.987 - 0.159i)T \)
97 \( 1 + (0.185 - 0.982i)T \)
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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

−22.87955305973521429899842944262, −20.96183224724788019369790336334, −20.82863216333481597124993769041, −19.976267784306247114055236054972, −19.062061957390079840818258819703, −18.064825171620089707650047445008, −17.78693268481552526519241794395, −17.01839461921842070159545480262, −16.19268498596417411948171204667, −14.981475984675411307820544506465, −14.15650135551254574183797019713, −13.57956679550505122947034525744, −12.44482529758858044533161883041, −11.57471737238883331410735034464, −10.656023176360945761377063940007, −9.49698510742671733730917562813, −8.966067849215691872476605036939, −7.90273576859853793078635508765, −7.38942190228912072329790342555, −6.34102579135877752731035780444, −5.568806673555763418357943133182, −4.36095454555220169538403821500, −2.497751347711635764598503621682, −1.67759354887669067030338799174, −1.00894002774887644548550188889, 1.303661744007419544593190707757, 2.57407460940955901110417990002, 3.10789559145492240821695551578, 4.39805142496876430687761066111, 5.72041198696519188166777904293, 6.41485583366381156698726296253, 7.918602467481631450592670919260, 8.69175383251499562033404358916, 9.10954545565841592892298597624, 10.47097823566353330236050911489, 10.730566168657209098752595217160, 11.388027370365345385703051625803, 12.75211307824755077818092431213, 13.805460442814544996992010030186, 14.73164185536333901225924860615, 15.429829445801194629371471811240, 16.31114942291247506184883569311, 17.20772836010033318084291899270, 17.863839336982659695700058636985, 18.640972776112742349676998830127, 19.48721047100350301764026837865, 20.41867948665368093759842234234, 21.36780407422481579185869310392, 21.47630400245729297912205488393, 22.24699422508295088351106706648

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