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Properties

Label 1-731-731.105-r0-0-0
Degree $1$
Conductor $731$
Sign $-0.853 + 0.520i$
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  + (0.846 + 0.532i)2-s + (−0.995 − 0.0933i)3-s + (0.433 + 0.900i)4-s + (−0.720 − 0.693i)5-s + (−0.793 − 0.608i)6-s + (−0.991 − 0.130i)7-s + (−0.111 + 0.993i)8-s + (0.982 + 0.185i)9-s + (−0.240 − 0.970i)10-s + (0.998 + 0.0560i)11-s + (−0.347 − 0.937i)12-s + (−0.149 − 0.988i)13-s + (−0.770 − 0.638i)14-s + (0.652 + 0.757i)15-s + (−0.623 + 0.781i)16-s + ⋯
L(s)  = 1  + (0.846 + 0.532i)2-s + (−0.995 − 0.0933i)3-s + (0.433 + 0.900i)4-s + (−0.720 − 0.693i)5-s + (−0.793 − 0.608i)6-s + (−0.991 − 0.130i)7-s + (−0.111 + 0.993i)8-s + (0.982 + 0.185i)9-s + (−0.240 − 0.970i)10-s + (0.998 + 0.0560i)11-s + (−0.347 − 0.937i)12-s + (−0.149 − 0.988i)13-s + (−0.770 − 0.638i)14-s + (0.652 + 0.757i)15-s + (−0.623 + 0.781i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.853 + 0.520i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ -0.853 + 0.520i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1853203594 + 0.6603865915i\)
\(L(\frac12)\) \(\approx\) \(0.1853203594 + 0.6603865915i\)
\(L(1)\) \(\approx\) \(0.8268109561 + 0.3125912128i\)
\(L(1)\) \(\approx\) \(0.8268109561 + 0.3125912128i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.846 + 0.532i)T \)
3 \( 1 + (-0.995 - 0.0933i)T \)
5 \( 1 + (-0.720 - 0.693i)T \)
7 \( 1 + (-0.991 - 0.130i)T \)
11 \( 1 + (0.998 + 0.0560i)T \)
13 \( 1 + (-0.149 - 0.988i)T \)
19 \( 1 + (-0.982 + 0.185i)T \)
23 \( 1 + (0.450 + 0.892i)T \)
29 \( 1 + (-0.638 + 0.770i)T \)
31 \( 1 + (0.937 - 0.347i)T \)
37 \( 1 + (0.130 + 0.991i)T \)
41 \( 1 + (-0.815 + 0.578i)T \)
47 \( 1 + (-0.433 - 0.900i)T \)
53 \( 1 + (-0.804 + 0.593i)T \)
59 \( 1 + (-0.993 + 0.111i)T \)
61 \( 1 + (-0.347 + 0.937i)T \)
67 \( 1 + (-0.826 - 0.563i)T \)
71 \( 1 + (-0.450 + 0.892i)T \)
73 \( 1 + (-0.240 + 0.970i)T \)
79 \( 1 + (0.130 - 0.991i)T \)
83 \( 1 + (-0.467 + 0.884i)T \)
89 \( 1 + (0.294 + 0.955i)T \)
97 \( 1 + (-0.745 + 0.666i)T \)
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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

−22.33948436166823367577734124673, −21.70113742947465556343818811971, −20.81167871967999637167827481339, −19.45962115383169216594134657464, −19.2129279142586356868615198528, −18.474417798654434679960547747396, −17.11263580442694948842362602153, −16.34743826612717677888257814369, −15.57175124883473123819678592595, −14.827373233557843932164490987455, −13.93074137480091821291397086013, −12.79596982385471067324176288511, −12.17930934910132885068393193902, −11.48378895525525022800923495505, −10.803442701170817697539451656711, −9.95840875745665643865617715251, −9.0457185127761723767020846226, −7.22453507324250683263540988781, −6.46150989095405682129556582268, −6.16713315383998335152117865887, −4.61657388038304901521513320311, −4.075464637210103071654344053160, −3.12075472986991855402955510927, −1.86490172884993216441870243151, −0.288753968753878637721843724695, 1.33454684695168275946095033721, 3.15498787619713986382216071000, 4.00523139699078481521339073545, 4.82167862382630241265097163051, 5.75197929974502412577883909844, 6.5454845033090121441416191209, 7.30148071157347519096382136179, 8.279526284685515539172166519568, 9.41563985259074020493972906161, 10.60015896064158452263910190235, 11.623164260881556999824313249933, 12.20471687068801995021530029222, 12.932858257546832454939142931506, 13.4631067290719128673914801093, 15.042367804823880346545968443167, 15.44139222710541511898775213554, 16.45578336872721678896073223199, 16.87171306428470478825347617973, 17.52245586469652144522062225676, 18.876205016459281101353053100104, 19.74988519955455886780797107218, 20.48740285506689174350075781038, 21.6243812232962848923081236729, 22.26681967239753541084123268011, 23.04832780884671384951942646118

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