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Properties

Label 2-731-17.2-c1-0-9
Degree $2$
Conductor $731$
Sign $0.891 - 0.453i$
Analytic cond. $5.83706$
Root an. cond. $2.41600$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.01 − 1.01i)2-s + (1.18 − 0.490i)3-s + 0.0504i·4-s + (0.326 + 0.788i)5-s + (−1.69 − 0.703i)6-s + (−1.15 + 2.78i)7-s + (−1.97 + 1.97i)8-s + (−0.957 + 0.957i)9-s + (0.467 − 1.12i)10-s + (−0.413 − 0.171i)11-s + (0.0247 + 0.0598i)12-s − 3.65i·13-s + (3.99 − 1.65i)14-s + (0.774 + 0.774i)15-s + 4.09·16-s + (−3.58 + 2.04i)17-s + ⋯
L(s)  = 1  + (−0.715 − 0.715i)2-s + (0.684 − 0.283i)3-s + 0.0252i·4-s + (0.146 + 0.352i)5-s + (−0.692 − 0.287i)6-s + (−0.436 + 1.05i)7-s + (−0.697 + 0.697i)8-s + (−0.319 + 0.319i)9-s + (0.147 − 0.357i)10-s + (−0.124 − 0.0516i)11-s + (0.00715 + 0.0172i)12-s − 1.01i·13-s + (1.06 − 0.441i)14-s + (0.199 + 0.199i)15-s + 1.02·16-s + (−0.868 + 0.494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.891 - 0.453i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.891 - 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.938969 + 0.225415i\)
\(L(\frac12)\) \(\approx\) \(0.938969 + 0.225415i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.58 - 2.04i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1.01 + 1.01i)T + 2iT^{2} \)
3 \( 1 + (-1.18 + 0.490i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.326 - 0.788i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.15 - 2.78i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.413 + 0.171i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 3.65iT - 13T^{2} \)
19 \( 1 + (-5.83 - 5.83i)T + 19iT^{2} \)
23 \( 1 + (-3.61 - 1.49i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.12 - 7.53i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (8.00 - 3.31i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-4.14 + 1.71i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.134 - 0.325i)T + (-28.9 - 28.9i)T^{2} \)
47 \( 1 + 4.34iT - 47T^{2} \)
53 \( 1 + (-3.03 - 3.03i)T + 53iT^{2} \)
59 \( 1 + (5.34 - 5.34i)T - 59iT^{2} \)
61 \( 1 + (1.91 - 4.62i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 2.10T + 67T^{2} \)
71 \( 1 + (3.53 - 1.46i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-0.651 - 1.57i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (5.99 + 2.48i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-6.80 - 6.80i)T + 83iT^{2} \)
89 \( 1 - 3.03iT - 89T^{2} \)
97 \( 1 + (3.06 + 7.39i)T + (-68.5 + 68.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.51552047393483027800686643451, −9.511046276127912433434453658798, −8.823789428458566643507939868233, −8.271943233412741295487559830660, −7.20122152670299205029904898430, −5.85094213615885522961547381269, −5.33693737431602433399016838707, −3.20101988193134183828446928419, −2.72530958173791448655321290521, −1.55327894814703095177263902459, 0.58777409868225661633890233660, 2.75536502751558171644942108981, 3.75609372821555579881013663679, 4.78284322014620108850041261014, 6.30343527959373240154740262772, 7.05495450029476658321441199618, 7.67722022788616628263604058376, 8.839918112343591020962809352958, 9.261431456895904015897830670375, 9.755297000806006801984765536786

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