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Properties

Label 1-731-731.343-r0-0-0
Degree $1$
Conductor $731$
Sign $-0.982 - 0.185i$
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 + (−0.923 − 0.382i)3-s − i·4-s + (0.382 − 0.923i)5-s + (0.923 − 0.382i)6-s + (0.382 + 0.923i)7-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (0.382 + 0.923i)10-s + (−0.923 + 0.382i)11-s + (−0.382 + 0.923i)12-s + i·13-s + (−0.923 − 0.382i)14-s + (−0.707 + 0.707i)15-s − 16-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.923 − 0.382i)3-s − i·4-s + (0.382 − 0.923i)5-s + (0.923 − 0.382i)6-s + (0.382 + 0.923i)7-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (0.382 + 0.923i)10-s + (−0.923 + 0.382i)11-s + (−0.382 + 0.923i)12-s + i·13-s + (−0.923 − 0.382i)14-s + (−0.707 + 0.707i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.982 - 0.185i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ -0.982 - 0.185i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009870837900 + 0.1056962556i\)
\(L(\frac12)\) \(\approx\) \(0.009870837900 + 0.1056962556i\)
\(L(1)\) \(\approx\) \(0.4712009229 + 0.1047677582i\)
\(L(1)\) \(\approx\) \(0.4712009229 + 0.1047677582i\)

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 \( 1 + (-0.923 - 0.382i)T \)
5 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + iT \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (-0.382 + 0.923i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.923 - 0.382i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + iT \)
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

−21.85961819352271435605302038673, −21.318957140598068361976654948544, −20.62096659123279629148634809516, −19.63185634588611335278790408486, −18.61027894193506919049564230848, −18.03438002078478681921262134503, −17.31994128540767945254281447164, −16.815404144841803049895478652033, −15.66532712520021759294503094788, −14.934250494041227945436072424230, −13.423246863076007352152096076244, −13.06322908290191387283605758688, −11.713810062079679706062045713588, −10.97266107722413208426505888461, −10.48803525880288902255324639187, −10.01927339604354856206622701735, −8.7784662759146809171325451176, −7.54520947123208265178578272627, −7.01011305896966381000613901596, −5.79121892395428233082836541107, −4.73099956564898070284971932245, −3.60779368786765577013631864118, −2.75286556374215345192224291341, −1.37658806822128161896444526668, −0.07367831133516557888296630388, 1.565750018204365801934691378, 2.07936987081495920070759534478, 4.524699644985619921997279465763, 5.22391122752179126482992447903, 5.80638213907380291338751356699, 6.784760473577761823276445159781, 7.73046723222292828920355027539, 8.66469022886843292279838930937, 9.336201535249588899179686420055, 10.4189155910169846629758092622, 11.18754450993521691891314309344, 12.28750013812980239848620869243, 12.87598869764047782951337021812, 13.98344856364830126456558510271, 15.01522820082031718969524566466, 15.922773858681459181304373973212, 16.56837258100841180269125913155, 17.20740343889108779205792500642, 18.025569966858030705120148875592, 18.6324689764485665857050956464, 19.284940867686242251338619781994, 20.64973649424935694455764621064, 21.24753317183408101771629082249, 22.27748289182110691826265188858, 23.29928731970026836432746995794

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