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Properties

Label 1-731-731.353-r0-0-0
Degree $1$
Conductor $731$
Sign $-0.350 - 0.936i$
Analytic cond. $3.39474$
Root an. cond. $3.39474$
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  + (0.222 − 0.974i)2-s + (−0.294 + 0.955i)3-s + (−0.900 − 0.433i)4-s + (−0.930 + 0.365i)5-s + (0.866 + 0.5i)6-s + (−0.866 + 0.5i)7-s + (−0.623 + 0.781i)8-s + (−0.826 − 0.563i)9-s + (0.149 + 0.988i)10-s + (0.433 + 0.900i)11-s + (0.680 − 0.733i)12-s + (−0.988 − 0.149i)13-s + (0.294 + 0.955i)14-s + (−0.0747 − 0.997i)15-s + (0.623 + 0.781i)16-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.294 + 0.955i)3-s + (−0.900 − 0.433i)4-s + (−0.930 + 0.365i)5-s + (0.866 + 0.5i)6-s + (−0.866 + 0.5i)7-s + (−0.623 + 0.781i)8-s + (−0.826 − 0.563i)9-s + (0.149 + 0.988i)10-s + (0.433 + 0.900i)11-s + (0.680 − 0.733i)12-s + (−0.988 − 0.149i)13-s + (0.294 + 0.955i)14-s + (−0.0747 − 0.997i)15-s + (0.623 + 0.781i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.350 - 0.936i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ -0.350 - 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1401544232 - 0.2020583082i\)
\(L(\frac12)\) \(\approx\) \(0.1401544232 - 0.2020583082i\)
\(L(1)\) \(\approx\) \(0.5719236741 - 0.04166335724i\)
\(L(1)\) \(\approx\) \(0.5719236741 - 0.04166335724i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.222 - 0.974i)T \)
3 \( 1 + (-0.294 + 0.955i)T \)
5 \( 1 + (-0.930 + 0.365i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.433 + 0.900i)T \)
13 \( 1 + (-0.988 - 0.149i)T \)
19 \( 1 + (-0.826 + 0.563i)T \)
23 \( 1 + (-0.997 - 0.0747i)T \)
29 \( 1 + (0.294 + 0.955i)T \)
31 \( 1 + (0.680 - 0.733i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.974 + 0.222i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.988 - 0.149i)T \)
59 \( 1 + (-0.623 - 0.781i)T \)
61 \( 1 + (0.680 + 0.733i)T \)
67 \( 1 + (0.826 - 0.563i)T \)
71 \( 1 + (-0.997 + 0.0747i)T \)
73 \( 1 + (0.149 - 0.988i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (-0.955 - 0.294i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (-0.433 - 0.900i)T \)
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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

−22.99292734663991516325517743394, −22.3814112783084934859484401998, −21.47086889583053915389980861116, −19.948317944952467016165943046519, −19.29937159348650289535975051470, −18.908638477334364586042862562084, −17.568650926509096776423943622990, −17.02626994686113422351962572956, −16.29326895687291901870188285009, −15.62378446611593467124273761645, −14.45770478962767117379240915925, −13.73524860234324283898831239981, −12.928676977420024587473806258929, −12.2420146068085552054185761084, −11.50284666484910992807617284036, −10.15362466754152141616023949010, −8.91574821631818530083060453976, −8.2114152119739074106448001526, −7.36678706981891761844399595954, −6.677772159695608858665655915858, −5.94349373749994455801946193765, −4.76647677731448357892169384512, −3.86311123463478974907270507566, −2.77664137085067654080964754990, −0.83101025049377385523088436312, 0.15742112096286613193648810210, 2.20425130384525534765105144594, 3.14786820744326744337602493109, 3.99676700913344463016424553185, 4.63681441512195853859283585744, 5.74196288989253999433749372135, 6.78463990350359164422489420757, 8.201336034730805491744904236312, 9.16262967585447541321968887809, 9.98630767420793470344401078947, 10.46198251513456001397188345055, 11.602448869520036878668728891348, 12.176991404721783481132748794103, 12.73106750937623704254416918390, 14.308121370240529625309496141596, 14.85091231481099591864125548190, 15.5588037834772393865721166553, 16.50301425701894109817851641344, 17.4853065783410138795029231587, 18.3764492445400185130456261911, 19.473955721303233157555914906899, 19.74384566159662416386893836207, 20.63462413298413408928791142351, 21.625606295747028779269632217193, 22.27207188004564352772734581389

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