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Properties

Label 1-693-693.248-r1-0-0
Degree $1$
Conductor $693$
Sign $-0.675 + 0.737i$
Analytic cond. $74.4731$
Root an. cond. $74.4731$
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  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (0.913 + 0.406i)13-s + (−0.104 + 0.994i)16-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.104 − 0.994i)20-s − 23-s + (0.309 + 0.951i)25-s + (0.669 + 0.743i)26-s + (0.669 + 0.743i)29-s + (0.104 + 0.994i)31-s + (−0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (0.913 + 0.406i)13-s + (−0.104 + 0.994i)16-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.104 − 0.994i)20-s − 23-s + (0.309 + 0.951i)25-s + (0.669 + 0.743i)26-s + (0.669 + 0.743i)29-s + (0.104 + 0.994i)31-s + (−0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.675 + 0.737i$
Analytic conductor: \(74.4731\)
Root analytic conductor: \(74.4731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (1:\ ),\ -0.675 + 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8838710812 + 2.009794108i\)
\(L(\frac12)\) \(\approx\) \(0.8838710812 + 2.009794108i\)
\(L(1)\) \(\approx\) \(1.393632717 + 0.5229194884i\)
\(L(1)\) \(\approx\) \(1.393632717 + 0.5229194884i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.913 + 0.406i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 - T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−22.24681208376395303897889187767, −21.44379353755693073429558574476, −20.62365580205257769462257670448, −19.78528006386336209300212162758, −19.10285160826972197191717484643, −18.42304466007895512102437681979, −17.21207155680809785126313281902, −15.99486716086458220727742324738, −15.47918588168277492639843780110, −14.68443711277443798466200807136, −13.93760027127476474834795392593, −12.903289529314136338393349209120, −12.25744454487306710486236307031, −11.25094114203464801644740089629, −10.73272030908713708989438128667, −9.86949948281457647021396561699, −8.37871983855929298826857870646, −7.61162688191896797709796640255, −6.33036950525714934782002524464, −5.93832651096578586246564968963, −4.36594196071017517082018252203, −3.88890829637594813719086292721, −2.88193438606886888686331976688, −1.79724184268279220453116676890, −0.36405123707584091155287281314, 1.27646905957850666063669520973, 2.69255716050029319478804993280, 3.77287974703499636017960002233, 4.475967908251458096558231691783, 5.35252529627822477064276010017, 6.45944576159616969762799889818, 7.24508938730460414571616142508, 8.310000243948046999515415392487, 8.83522211000452720992665061119, 10.3906853363738100308541352754, 11.45470059202247892800948016288, 11.97549592808137965466687784917, 12.87255283535570998573073841918, 13.6556016157116012948156469943, 14.49077616991378020896237159196, 15.47048627912846480972674381318, 16.09218625026088583756381177410, 16.63380339489836929627208172817, 17.71234097758715750062650824624, 18.75109880283773294650643429849, 19.835145427753134682335887342800, 20.42171757284220431378625384543, 21.2177310631016528380813492847, 22.02132892400304603571421830842, 22.99581762836283901980203159297

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