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Properties

Label 1-69-69.62-r1-0-0
Degree $1$
Conductor $69$
Sign $0.914 + 0.404i$
Analytic cond. $7.41507$
Root an. cond. $7.41507$
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  + (0.142 + 0.989i)2-s + (−0.959 + 0.281i)4-s + (0.654 − 0.755i)5-s + (0.841 − 0.540i)7-s + (−0.415 − 0.909i)8-s + (0.841 + 0.540i)10-s + (0.142 − 0.989i)11-s + (0.841 + 0.540i)13-s + (0.654 + 0.755i)14-s + (0.841 − 0.540i)16-s + (0.959 + 0.281i)17-s + (−0.959 + 0.281i)19-s + (−0.415 + 0.909i)20-s + 22-s + (−0.142 − 0.989i)25-s + (−0.415 + 0.909i)26-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)2-s + (−0.959 + 0.281i)4-s + (0.654 − 0.755i)5-s + (0.841 − 0.540i)7-s + (−0.415 − 0.909i)8-s + (0.841 + 0.540i)10-s + (0.142 − 0.989i)11-s + (0.841 + 0.540i)13-s + (0.654 + 0.755i)14-s + (0.841 − 0.540i)16-s + (0.959 + 0.281i)17-s + (−0.959 + 0.281i)19-s + (−0.415 + 0.909i)20-s + 22-s + (−0.142 − 0.989i)25-s + (−0.415 + 0.909i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.914 + 0.404i$
Analytic conductor: \(7.41507\)
Root analytic conductor: \(7.41507\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 69,\ (1:\ ),\ 0.914 + 0.404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.866454197 + 0.3945266845i\)
\(L(\frac12)\) \(\approx\) \(1.866454197 + 0.3945266845i\)
\(L(1)\) \(\approx\) \(1.278054093 + 0.3325360116i\)
\(L(1)\) \(\approx\) \(1.278054093 + 0.3325360116i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.142 + 0.989i)T \)
5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (0.841 - 0.540i)T \)
11 \( 1 + (0.142 - 0.989i)T \)
13 \( 1 + (0.841 + 0.540i)T \)
17 \( 1 + (0.959 + 0.281i)T \)
19 \( 1 + (-0.959 + 0.281i)T \)
29 \( 1 + (0.959 + 0.281i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (-0.654 - 0.755i)T \)
41 \( 1 + (0.654 - 0.755i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.841 + 0.540i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (0.415 + 0.909i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (-0.959 + 0.281i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (0.654 + 0.755i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−31.11502514216777583670866260731, −30.36769956193914399653244321212, −29.58146037191411701417713552197, −28.1628772417505416653411303463, −27.593848063826330870314724063629, −26.117403065345496775423845793809, −25.07560778283130896942258106282, −23.3894231080556825222442222107, −22.51744318824045906605779355363, −21.3217078374173922200699247685, −20.69744829897548164986541152244, −19.14871271678705057376008537758, −18.12238774917979865944017843596, −17.44585933062640840451712412938, −15.13636242238605123500603816486, −14.29164178945214000122888552096, −13.02480538348122589652654030834, −11.73645701036607403766832484815, −10.63283692803735190672451163535, −9.58133421393508575181185798995, −8.10392010855262101208130412879, −6.079874528213843794984437209933, −4.679088108887632028373671651816, −2.90277147030864019960691660893, −1.616964021996128329592665960060, 1.14637279172890918341549447822, 3.94742514485253257020278077870, 5.27158071707526897012188628328, 6.416601459670494923902922097971, 8.12241314210404600925653392981, 8.91403354693090072723697954303, 10.55466371234378343679850656155, 12.37618846557227332907971559210, 13.74152604938944844057319276912, 14.30988162722948778276053294135, 16.02845143613447761584026797208, 16.8813076745988220387243045382, 17.77583349043345991378828444834, 19.110960861977109756433741233535, 20.995375516782376788535757680494, 21.52234750624693352380605482046, 23.28177675000323371157125708855, 24.0056100853114793787460306595, 24.990812837243420767164904472581, 25.99247708309127300942019898689, 27.20415904138823323728844767038, 28.09425266749580764288728460253, 29.59617549680169795508216900701, 30.756568813317556501644448247211, 32.02551036258767987955263500398

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