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Properties

Label 1-731-731.252-r0-0-0
Degree $1$
Conductor $731$
Sign $0.883 - 0.468i$
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  + (−0.707 + 0.707i)2-s + (−0.130 − 0.991i)3-s − i·4-s + (0.991 − 0.130i)5-s + (0.793 + 0.608i)6-s + (0.991 + 0.130i)7-s + (0.707 + 0.707i)8-s + (−0.965 + 0.258i)9-s + (−0.608 + 0.793i)10-s + (0.923 − 0.382i)11-s + (−0.991 + 0.130i)12-s + (−0.866 − 0.5i)13-s + (−0.793 + 0.608i)14-s + (−0.258 − 0.965i)15-s − 16-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.130 − 0.991i)3-s − i·4-s + (0.991 − 0.130i)5-s + (0.793 + 0.608i)6-s + (0.991 + 0.130i)7-s + (0.707 + 0.707i)8-s + (−0.965 + 0.258i)9-s + (−0.608 + 0.793i)10-s + (0.923 − 0.382i)11-s + (−0.991 + 0.130i)12-s + (−0.866 − 0.5i)13-s + (−0.793 + 0.608i)14-s + (−0.258 − 0.965i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.883 - 0.468i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (252, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.883 - 0.468i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.308153455 - 0.3254952259i\)
\(L(\frac12)\) \(\approx\) \(1.308153455 - 0.3254952259i\)
\(L(1)\) \(\approx\) \(0.9979734208 - 0.09405032654i\)
\(L(1)\) \(\approx\) \(0.9979734208 - 0.09405032654i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-0.130 - 0.991i)T \)
5 \( 1 + (0.991 - 0.130i)T \)
7 \( 1 + (0.991 + 0.130i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.965 + 0.258i)T \)
23 \( 1 + (0.793 + 0.608i)T \)
29 \( 1 + (0.608 + 0.793i)T \)
31 \( 1 + (-0.130 - 0.991i)T \)
37 \( 1 + (-0.130 - 0.991i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.965 - 0.258i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.991 - 0.130i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.793 + 0.608i)T \)
73 \( 1 + (-0.608 - 0.793i)T \)
79 \( 1 + (-0.130 + 0.991i)T \)
83 \( 1 + (0.965 + 0.258i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.382 + 0.923i)T \)
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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

−22.12547399324472491084427096733, −21.69193205987632156446503053465, −20.92482320266619693812484319672, −20.354111662518905698892838194, −19.525172424309997432499514391429, −18.39425224037342608104342548435, −17.46465597517092784013818900676, −17.21243384675334193386303579897, −16.43409068123378014374121045941, −15.19832665383842523073870849222, −14.309151942515416674244729840052, −13.707281791346380468680075614294, −12.22148198496380156000683096754, −11.65892852061567808079602498261, −10.70953232089654557190217344422, −10.08327296404588819263952959456, −9.251310825817253454766252832341, −8.77024829280809392929602140102, −7.46305045247888676014360987017, −6.466950593837847864479213667439, −5.015194331201297511094473809938, −4.4962986969266018259647052624, −3.21838538815251178763692817730, −2.23775113578687007995817076373, −1.19749364750546576562841978781, 1.04872318984339168702830827712, 1.64495120739097486191597239332, 2.77208178468196815479349978741, 4.86109077142012293566297405486, 5.5563771805698344115285156943, 6.3012183933024291593475141448, 7.29329054495047934755822046583, 7.93795933521461340253492381645, 8.94994062978511283643890936012, 9.577356804485433606674093236870, 10.80822150121365058417595115172, 11.53580395469853082273188141083, 12.59997208988530766923056453537, 13.68065992744530172697160827584, 14.32958337488407686937325980637, 14.81678129848384694159410278037, 16.24534278330606753942672967642, 17.174994444340269856756235813403, 17.51148814174916804521233184918, 18.15948093161888278083835332306, 18.977392818977445023264926506433, 19.84306987492084958244383491827, 20.57614171725748160478973139905, 21.806371038675425720324906761, 22.585487424855787322226135476751

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