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Properties

Label 2-731-17.2-c1-0-61
Degree $2$
Conductor $731$
Sign $-0.477 - 0.878i$
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  + (−0.692 − 0.692i)2-s + (0.747 − 0.309i)3-s − 1.04i·4-s + (−1.63 − 3.93i)5-s + (−0.732 − 0.303i)6-s + (−0.263 + 0.636i)7-s + (−2.10 + 2.10i)8-s + (−1.65 + 1.65i)9-s + (−1.59 + 3.85i)10-s + (−1.48 − 0.617i)11-s + (−0.322 − 0.778i)12-s − 2.88i·13-s + (0.623 − 0.258i)14-s + (−2.43 − 2.43i)15-s + 0.835·16-s + (2.13 + 3.52i)17-s + ⋯
L(s)  = 1  + (−0.489 − 0.489i)2-s + (0.431 − 0.178i)3-s − 0.520i·4-s + (−0.728 − 1.75i)5-s + (−0.299 − 0.123i)6-s + (−0.0997 + 0.240i)7-s + (−0.744 + 0.744i)8-s + (−0.552 + 0.552i)9-s + (−0.504 + 1.21i)10-s + (−0.449 − 0.186i)11-s + (−0.0930 − 0.224i)12-s − 0.800i·13-s + (0.166 − 0.0690i)14-s + (−0.629 − 0.629i)15-s + 0.208·16-s + (0.517 + 0.855i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.477 - 0.878i$
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: \((2,\ 731,\ (\ :1/2),\ -0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.220473 + 0.370571i\)
\(L(\frac12)\) \(\approx\) \(0.220473 + 0.370571i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-2.13 - 3.52i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (0.692 + 0.692i)T + 2iT^{2} \)
3 \( 1 + (-0.747 + 0.309i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.63 + 3.93i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.263 - 0.636i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.48 + 0.617i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 2.88iT - 13T^{2} \)
19 \( 1 + (-0.262 - 0.262i)T + 19iT^{2} \)
23 \( 1 + (1.85 + 0.769i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (2.34 + 5.66i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-8.67 + 3.59i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (0.0896 - 0.0371i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.640 - 1.54i)T + (-28.9 - 28.9i)T^{2} \)
47 \( 1 + 9.07iT - 47T^{2} \)
53 \( 1 + (4.98 + 4.98i)T + 53iT^{2} \)
59 \( 1 + (7.71 - 7.71i)T - 59iT^{2} \)
61 \( 1 + (-3.09 + 7.46i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 + (8.62 - 3.57i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-4.79 - 11.5i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (8.64 + 3.57i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (10.0 + 10.0i)T + 83iT^{2} \)
89 \( 1 + 1.65iT - 89T^{2} \)
97 \( 1 + (-4.39 - 10.6i)T + (-68.5 + 68.5i)T^{2} \)
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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

−9.745816062521620713172197776622, −8.852535492400841159373610095830, −8.188664801969054174303656553146, −7.88622218056377239094728742405, −5.88256846232780779807114826895, −5.36774515411135794118749143323, −4.28991673210117600787537465412, −2.83485752326541179053263973423, −1.53575496692319676560719804768, −0.24561809169936619353151368140, 2.80809198866585908182275141304, 3.24907740903759607282606216640, 4.27183439770270915929319320351, 6.11166697095230681499167610093, 6.90196992518054472173743736552, 7.45708514511273586384138571217, 8.205746032452859677618754666105, 9.183866189710723479693627477732, 9.977459449153049337555747017262, 10.88040053785531866679550469624

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