Properties

Label 1-693-693.515-r1-0-0
Degree 11
Conductor 693693
Sign 0.2920.956i-0.292 - 0.956i
Analytic cond. 74.473174.4731
Root an. cond. 74.473174.4731
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(693s/2ΓR(s+1)L(s)=((0.2920.956i)Λ(1s)\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}
Λ(s)=(693s/2ΓR(s+1)L(s)=((0.2920.956i)Λ(1s)\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: 11
Conductor: 693693    =    327113^{2} \cdot 7 \cdot 11
Sign: 0.2920.956i-0.292 - 0.956i
Analytic conductor: 74.473174.4731
Root analytic conductor: 74.473174.4731
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ693(515,)\chi_{693} (515, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 693, (1: ), 0.2920.956i)(1,\ 693,\ (1:\ ),\ -0.292 - 0.956i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.17766837980.2401457858i0.1776683798 - 0.2401457858i
L(12)L(\frac12) \approx 0.17766837980.2401457858i0.1776683798 - 0.2401457858i
L(1)L(1) \approx 1.102696826+0.3859584811i1.102696826 + 0.3859584811i
L(1)L(1) \approx 1.102696826+0.3859584811i1.102696826 + 0.3859584811i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
7 1 1
11 1 1
good2 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
5 1+(0.9130.406i)T 1 + (-0.913 - 0.406i)T
13 1+(0.9130.406i)T 1 + (0.913 - 0.406i)T
17 1+(0.9130.406i)T 1 + (-0.913 - 0.406i)T
19 1+(0.669+0.743i)T 1 + (0.669 + 0.743i)T
23 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
29 1+(0.669+0.743i)T 1 + (-0.669 + 0.743i)T
31 1+(0.8090.587i)T 1 + (-0.809 - 0.587i)T
37 1+(0.9780.207i)T 1 + (-0.978 - 0.207i)T
41 1+(0.6690.743i)T 1 + (-0.669 - 0.743i)T
43 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
47 1+(0.309+0.951i)T 1 + (-0.309 + 0.951i)T
53 1+(0.1040.994i)T 1 + (0.104 - 0.994i)T
59 1+(0.3090.951i)T 1 + (-0.309 - 0.951i)T
61 1+(0.809+0.587i)T 1 + (-0.809 + 0.587i)T
67 1+T 1 + T
71 1+(0.8090.587i)T 1 + (0.809 - 0.587i)T
73 1+(0.6690.743i)T 1 + (0.669 - 0.743i)T
79 1+(0.8090.587i)T 1 + (-0.809 - 0.587i)T
83 1+(0.9130.406i)T 1 + (-0.913 - 0.406i)T
89 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
97 1+(0.104+0.994i)T 1 + (-0.104 + 0.994i)T
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   L(s)=p (1αpps)1L(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 ZZ-function along the critical line