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Properties

Label 1-693-693.515-r1-0-0
Degree $1$
Conductor $693$
Sign $-0.292 - 0.956i$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.913 − 0.406i)5-s + (−0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (0.913 − 0.406i)13-s + (−0.809 + 0.587i)16-s + (−0.913 − 0.406i)17-s + (0.669 + 0.743i)19-s + (0.104 − 0.994i)20-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)25-s + (0.978 + 0.207i)26-s + (−0.669 + 0.743i)29-s + (−0.809 − 0.587i)31-s − 32-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.913 − 0.406i)5-s + (−0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (0.913 − 0.406i)13-s + (−0.809 + 0.587i)16-s + (−0.913 − 0.406i)17-s + (0.669 + 0.743i)19-s + (0.104 − 0.994i)20-s + (0.5 + 0.866i)23-s + (0.669 + 0.743i)25-s + (0.978 + 0.207i)26-s + (−0.669 + 0.743i)29-s + (−0.809 − 0.587i)31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.292 - 0.956i$
Analytic conductor: \(74.4731\)
Root analytic conductor: \(74.4731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (1:\ ),\ -0.292 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1776683798 - 0.2401457858i\)
\(L(\frac12)\) \(\approx\) \(0.1776683798 - 0.2401457858i\)
\(L(1)\) \(\approx\) \(1.102696826 + 0.3859584811i\)
\(L(1)\) \(\approx\) \(1.102696826 + 0.3859584811i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (0.913 - 0.406i)T \)
17 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (-0.669 - 0.743i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.669 - 0.743i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
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

−22.76791907870489526302961723513, −21.9497826065951902902700557827, −21.18520341791217421646871365252, −20.11910622432803916092780578300, −19.7932074123603782200089365312, −18.67172781741168357336128989783, −18.288019365657169115699650491, −16.79392184045739536211646262046, −15.747432393171253407942683144545, −15.306399532003701606536101573802, −14.39412892764934442880711012431, −13.51518613955657744094127035364, −12.76099006770254475856783034263, −11.737555337346956292169856095893, −11.163698112677961319956961971017, −10.51915495142643927029700701796, −9.278551073421319326939583640795, −8.34024268106344307890634101659, −7.00396210979749942393843162655, −6.45731622801961386490989473636, −5.180384952435514767507244238293, −4.2470463026254763410044078954, −3.5025420083424202417334839883, −2.54457808998680144333159263289, −1.27839409807207971673802284971, 0.05148222250034412616235393522, 1.702387595898033167543379502747, 3.32351515889929410447284803036, 3.76804180876861193495933096285, 4.94937380341285469316748157190, 5.63357318110930093521932452293, 6.85406368826913655846764563200, 7.57674122268196415417176382198, 8.44043334649705840395794744893, 9.22054147654271470695931382069, 10.86855889524570617570518586748, 11.491726373269862205753390368582, 12.378996598376016262461010508888, 13.14626058120251198766047256339, 13.89741350818404238945418852251, 14.97041761184928592879861349283, 15.65145235119978053583325077053, 16.19245337831187384444185279174, 17.06461071237603093544459249763, 18.03325753254729412040742294874, 18.95881209440797909257705610929, 20.28190029017798997754258386791, 20.43348249172407834261639745098, 21.5331120860216442252766529132, 22.58944734650220453903558889873

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