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Properties

Label 1-739-739.738-r1-0-0
Degree $1$
Conductor $739$
Sign $1$
Analytic cond. $79.4165$
Root an. cond. $79.4165$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 11-s − 12-s − 13-s + 14-s − 15-s + 16-s + 17-s − 18-s − 19-s + 20-s + 21-s − 22-s − 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 11-s − 12-s − 13-s + 14-s − 15-s + 16-s + 17-s − 18-s − 19-s + 20-s + 21-s − 22-s − 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 739 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 739 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(739\)
Sign: $1$
Analytic conductor: \(79.4165\)
Root analytic conductor: \(79.4165\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{739} (738, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 739,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8543487994\)
\(L(\frac12)\) \(\approx\) \(0.8543487994\)
\(L(1)\) \(\approx\) \(0.5778267693\)
\(L(1)\) \(\approx\) \(0.5778267693\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad739 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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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

−22.15383706953364843683577198210, −21.57892329603615968734388028490, −20.69335883533223481363715884664, −19.50996022780220914202437237728, −19.03559480140144393307955717961, −18.06405253777672270886584595494, −17.32042407776335982324384262737, −16.77002474362635043063752105391, −16.295484580430605749862849884986, −15.12288399686172235754564103839, −14.20586486399319898810776065483, −12.78184521351090083673181094474, −12.343513342157929709396941046526, −11.35574198701189840347809061798, −10.35767131750435878989628473185, −9.71183663509992183183118240349, −9.32802840705322774322603367170, −7.85031013807199415269007109191, −6.789251115712917238052730978852, −6.24231069610728597327278072915, −5.56950553390567582366266535822, −4.100839531078580143820717817606, −2.68497131744157144917594305070, −1.63486576112972507444418921867, −0.56953475733824429892847871648, 0.56953475733824429892847871648, 1.63486576112972507444418921867, 2.68497131744157144917594305070, 4.100839531078580143820717817606, 5.56950553390567582366266535822, 6.24231069610728597327278072915, 6.789251115712917238052730978852, 7.85031013807199415269007109191, 9.32802840705322774322603367170, 9.71183663509992183183118240349, 10.35767131750435878989628473185, 11.35574198701189840347809061798, 12.343513342157929709396941046526, 12.78184521351090083673181094474, 14.20586486399319898810776065483, 15.12288399686172235754564103839, 16.295484580430605749862849884986, 16.77002474362635043063752105391, 17.32042407776335982324384262737, 18.06405253777672270886584595494, 19.03559480140144393307955717961, 19.50996022780220914202437237728, 20.69335883533223481363715884664, 21.57892329603615968734388028490, 22.15383706953364843683577198210

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