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Properties

Label 1-731-731.108-r0-0-0
Degree $1$
Conductor $731$
Sign $-0.703 - 0.710i$
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  + (0.111 − 0.993i)2-s + (−0.276 + 0.960i)3-s + (−0.974 − 0.222i)4-s + (−0.745 − 0.666i)5-s + (0.923 + 0.382i)6-s + (0.382 − 0.923i)7-s + (−0.330 + 0.943i)8-s + (−0.846 − 0.532i)9-s + (−0.745 + 0.666i)10-s + (−0.167 + 0.985i)11-s + (0.483 − 0.875i)12-s + (0.433 + 0.900i)13-s + (−0.875 − 0.483i)14-s + (0.846 − 0.532i)15-s + (0.900 + 0.433i)16-s + ⋯
L(s)  = 1  + (0.111 − 0.993i)2-s + (−0.276 + 0.960i)3-s + (−0.974 − 0.222i)4-s + (−0.745 − 0.666i)5-s + (0.923 + 0.382i)6-s + (0.382 − 0.923i)7-s + (−0.330 + 0.943i)8-s + (−0.846 − 0.532i)9-s + (−0.745 + 0.666i)10-s + (−0.167 + 0.985i)11-s + (0.483 − 0.875i)12-s + (0.433 + 0.900i)13-s + (−0.875 − 0.483i)14-s + (0.846 − 0.532i)15-s + (0.900 + 0.433i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.703 - 0.710i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ -0.703 - 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2908888456 - 0.6973490751i\)
\(L(\frac12)\) \(\approx\) \(0.2908888456 - 0.6973490751i\)
\(L(1)\) \(\approx\) \(0.6873015054 - 0.3524634486i\)
\(L(1)\) \(\approx\) \(0.6873015054 - 0.3524634486i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.111 - 0.993i)T \)
3 \( 1 + (-0.276 + 0.960i)T \)
5 \( 1 + (-0.745 - 0.666i)T \)
7 \( 1 + (0.382 - 0.923i)T \)
11 \( 1 + (-0.167 + 0.985i)T \)
13 \( 1 + (0.433 + 0.900i)T \)
19 \( 1 + (0.846 - 0.532i)T \)
23 \( 1 + (0.167 - 0.985i)T \)
29 \( 1 + (0.483 - 0.875i)T \)
31 \( 1 + (-0.875 - 0.483i)T \)
37 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (-0.960 + 0.276i)T \)
47 \( 1 + (0.974 + 0.222i)T \)
53 \( 1 + (-0.330 - 0.943i)T \)
59 \( 1 + (0.943 - 0.330i)T \)
61 \( 1 + (0.483 + 0.875i)T \)
67 \( 1 + (0.222 - 0.974i)T \)
71 \( 1 + (-0.167 - 0.985i)T \)
73 \( 1 + (-0.745 - 0.666i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (-0.993 + 0.111i)T \)
89 \( 1 + (-0.781 - 0.623i)T \)
97 \( 1 + (0.815 - 0.578i)T \)
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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

−23.08766672528106288695702433445, −22.21349794047642244771470169397, −21.69564305214250328404655867309, −20.22197099234716177197169441589, −19.11156564984255883769643980346, −18.58970119434273039229265668079, −18.05847100650419005654424709296, −17.269419488158820031590528247519, −16.02277737496233210021489416595, −15.66170387744644739271104781021, −14.54320356247417579224709788217, −13.99200458961866263627645556579, −12.999528580952244451915217080480, −12.16525080991816026215710653018, −11.41155162749539514295051904887, −10.438801396759361713518647798820, −8.83502860167177499319054834193, −8.28307357876521385735233218167, −7.51881899306882676527075671809, −6.77098777877958425467302933555, −5.578075377273562183477577828482, −5.4284369323922591806095528040, −3.61539028823510098712762488210, −2.88081871317306941478680078788, −1.159474132204010414724613315544, 0.42633042494145660667646143337, 1.70697717196595656393618014017, 3.19279440486919095626678112666, 4.21722752147119782803630907735, 4.49651032438958197175200678884, 5.37236290863688369553215559623, 6.95201355555259653391670752611, 8.16875293346832200665011684603, 9.00384201158698662589506872456, 9.82252933880014834762979432013, 10.60055833555377220762310845202, 11.47680406087599785248170917792, 11.928150418748618435586923013166, 13.00497927970138800827149639112, 13.93450973430629726016902933644, 14.78471633010455872671955712561, 15.6800451765272862429900155577, 16.63712218436203986195399800617, 17.27730753862652247229122134590, 18.16899233603317121908706650002, 19.29871830972327699521774881341, 20.16024475655719991853850288202, 20.61614539345288391125529403720, 21.06869432874561171279674849450, 22.250253297219063547603758023069

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