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Properties

Label 2-731-17.16-c1-0-42
Degree $2$
Conductor $731$
Sign $-0.215 + 0.976i$
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  − 2.48·2-s + 0.676i·3-s + 4.16·4-s − 3.88i·5-s − 1.67i·6-s − 2.90i·7-s − 5.37·8-s + 2.54·9-s + 9.65i·10-s − 3.51i·11-s + 2.81i·12-s + 6.64·13-s + 7.22i·14-s + 2.63·15-s + 5.02·16-s + (0.886 − 4.02i)17-s + ⋯
L(s)  = 1  − 1.75·2-s + 0.390i·3-s + 2.08·4-s − 1.73i·5-s − 0.685i·6-s − 1.09i·7-s − 1.90·8-s + 0.847·9-s + 3.05i·10-s − 1.05i·11-s + 0.813i·12-s + 1.84·13-s + 1.93i·14-s + 0.679·15-s + 1.25·16-s + (0.215 − 0.976i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.215 + 0.976i$
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: \((2,\ 731,\ (\ :1/2),\ -0.215 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.487149 - 0.606135i\)
\(L(\frac12)\) \(\approx\) \(0.487149 - 0.606135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.886 + 4.02i)T \)
43 \( 1 + T \)
good2 \( 1 + 2.48T + 2T^{2} \)
3 \( 1 - 0.676iT - 3T^{2} \)
5 \( 1 + 3.88iT - 5T^{2} \)
7 \( 1 + 2.90iT - 7T^{2} \)
11 \( 1 + 3.51iT - 11T^{2} \)
13 \( 1 - 6.64T + 13T^{2} \)
19 \( 1 - 1.34T + 19T^{2} \)
23 \( 1 - 0.0542iT - 23T^{2} \)
29 \( 1 - 0.155iT - 29T^{2} \)
31 \( 1 - 6.18iT - 31T^{2} \)
37 \( 1 + 3.55iT - 37T^{2} \)
41 \( 1 - 8.28iT - 41T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 - 0.329T + 59T^{2} \)
61 \( 1 - 7.36iT - 61T^{2} \)
67 \( 1 - 7.69T + 67T^{2} \)
71 \( 1 - 0.947iT - 71T^{2} \)
73 \( 1 + 8.32iT - 73T^{2} \)
79 \( 1 - 12.9iT - 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 3.66iT - 97T^{2} \)
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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

−9.908282750618937273480666549802, −9.193171899973416215593264726591, −8.612011448609932200090026880224, −7.937191058090508158324216228394, −7.02484170632274044573623952581, −5.86693610433832334686010739235, −4.60899233475942847235847974745, −3.53773232062433479704723139032, −1.27624236652387231029420312413, −0.889942784730645360441376438872, 1.63554162931649482319598093483, 2.42883980065755701989277231407, 3.75150352280505378377329735470, 6.02146013428455576239391952915, 6.51269134292789747208137355175, 7.36250663084902780078731978299, 8.000901400095367465447875513837, 8.963181080601840264213420089340, 9.836922321428225822514318097696, 10.46595097718473848456258086424

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