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Properties

Label 2-731-17.16-c1-0-34
Degree $2$
Conductor $731$
Sign $0.378 + 0.925i$
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  + 0.366·2-s − 1.80i·3-s − 1.86·4-s + 2.01i·5-s − 0.659i·6-s + 2.54i·7-s − 1.41·8-s − 0.241·9-s + 0.737i·10-s − 5.65i·11-s + 3.35i·12-s − 0.166·13-s + 0.931i·14-s + 3.62·15-s + 3.21·16-s + (−1.56 − 3.81i)17-s + ⋯
L(s)  = 1  + 0.258·2-s − 1.03i·3-s − 0.932·4-s + 0.901i·5-s − 0.269i·6-s + 0.961i·7-s − 0.500·8-s − 0.0803·9-s + 0.233i·10-s − 1.70i·11-s + 0.969i·12-s − 0.0461·13-s + 0.249i·14-s + 0.936·15-s + 0.803·16-s + (−0.378 − 0.925i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.378 + 0.925i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (560, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.378 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11682 - 0.749492i\)
\(L(\frac12)\) \(\approx\) \(1.11682 - 0.749492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (1.56 + 3.81i)T \)
43 \( 1 + T \)
good2 \( 1 - 0.366T + 2T^{2} \)
3 \( 1 + 1.80iT - 3T^{2} \)
5 \( 1 - 2.01iT - 5T^{2} \)
7 \( 1 - 2.54iT - 7T^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 + 0.166T + 13T^{2} \)
19 \( 1 - 8.12T + 19T^{2} \)
23 \( 1 + 8.66iT - 23T^{2} \)
29 \( 1 - 3.73iT - 29T^{2} \)
31 \( 1 - 10.1iT - 31T^{2} \)
37 \( 1 + 7.40iT - 37T^{2} \)
41 \( 1 + 5.10iT - 41T^{2} \)
47 \( 1 - 7.92T + 47T^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
59 \( 1 - 6.96T + 59T^{2} \)
61 \( 1 - 0.320iT - 61T^{2} \)
67 \( 1 + 2.61T + 67T^{2} \)
71 \( 1 + 1.24iT - 71T^{2} \)
73 \( 1 + 3.36iT - 73T^{2} \)
79 \( 1 - 0.324iT - 79T^{2} \)
83 \( 1 + 8.16T + 83T^{2} \)
89 \( 1 - 9.69T + 89T^{2} \)
97 \( 1 - 0.511iT - 97T^{2} \)
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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

−10.30891144112979575619140509076, −9.037147294060429782469554257405, −8.653644074679321928728433900134, −7.53723967162277691277359333159, −6.68146892296767266734435646997, −5.81275166855991693760001485763, −4.98654777530764266457362853860, −3.37054758161554874756620966025, −2.66820421954184943171671964100, −0.78371856241642256739254911236, 1.30672803683561904039528131228, 3.53542071794327058486156110870, 4.34052232420703135009915090910, 4.75846149348214786989915916575, 5.66368916444284413557305756268, 7.27243935403384462135487189692, 7.985003081348623443881160560840, 9.262005135611172879974934447904, 9.720640799732109659795789516452, 10.07808535864599888907099308013

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