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Properties

Label 2-731-17.16-c1-0-57
Degree $2$
Conductor $731$
Sign $-0.164 - 0.986i$
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  − 2.34·2-s − 2.97i·3-s + 3.51·4-s − 3.06i·5-s + 6.97i·6-s − 3.33i·7-s − 3.54·8-s − 5.82·9-s + 7.20i·10-s + 2.35i·11-s − 10.4i·12-s + 0.866·13-s + 7.83i·14-s − 9.11·15-s + 1.30·16-s + (0.679 + 4.06i)17-s + ⋯
L(s)  = 1  − 1.66·2-s − 1.71i·3-s + 1.75·4-s − 1.37i·5-s + 2.84i·6-s − 1.26i·7-s − 1.25·8-s − 1.94·9-s + 2.27i·10-s + 0.709i·11-s − 3.01i·12-s + 0.240·13-s + 2.09i·14-s − 2.35·15-s + 0.327·16-s + (0.164 + 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.164 - 0.986i$
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: \((2,\ 731,\ (\ :1/2),\ -0.164 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.225012 + 0.265711i\)
\(L(\frac12)\) \(\approx\) \(0.225012 + 0.265711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.679 - 4.06i)T \)
43 \( 1 + T \)
good2 \( 1 + 2.34T + 2T^{2} \)
3 \( 1 + 2.97iT - 3T^{2} \)
5 \( 1 + 3.06iT - 5T^{2} \)
7 \( 1 + 3.33iT - 7T^{2} \)
11 \( 1 - 2.35iT - 11T^{2} \)
13 \( 1 - 0.866T + 13T^{2} \)
19 \( 1 + 5.87T + 19T^{2} \)
23 \( 1 - 2.76iT - 23T^{2} \)
29 \( 1 - 9.99iT - 29T^{2} \)
31 \( 1 + 6.56iT - 31T^{2} \)
37 \( 1 + 7.92iT - 37T^{2} \)
41 \( 1 + 11.5iT - 41T^{2} \)
47 \( 1 + 5.66T + 47T^{2} \)
53 \( 1 - 6.00T + 53T^{2} \)
59 \( 1 + 2.89T + 59T^{2} \)
61 \( 1 + 9.44iT - 61T^{2} \)
67 \( 1 - 2.18T + 67T^{2} \)
71 \( 1 - 1.24iT - 71T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 - 8.70iT - 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + 5.40T + 89T^{2} \)
97 \( 1 + 14.0iT - 97T^{2} \)
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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

−9.530564084087665902598018057119, −8.557281261601235675392279817382, −8.234487492962154561599388626596, −7.25517408210929092361690494816, −6.90670052130424628149663314221, −5.68634081825323100050634755583, −4.11734836580677039291841199842, −1.97435620262933673719158354095, −1.34922097990627913939270377665, −0.31978659210633484381385289660, 2.51037156291726709637763940994, 3.11846983810343307536106062199, 4.62838039430725209317495813512, 5.98692469028577390643511268327, 6.62844538022478150651079547716, 8.081214402049058527230292545521, 8.668080188704331974553060413004, 9.398144872933103468319737648977, 10.12946304452425868339092843025, 10.60034020207979778484602808306

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