Properties

Label 2-731-17.2-c1-0-1
Degree $2$
Conductor $731$
Sign $-0.913 - 0.407i$
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  + (−1.14 − 1.14i)2-s + (−2.60 + 1.07i)3-s + 0.606i·4-s + (0.611 + 1.47i)5-s + (4.20 + 1.74i)6-s + (−0.771 + 1.86i)7-s + (−1.59 + 1.59i)8-s + (3.50 − 3.50i)9-s + (0.987 − 2.38i)10-s + (−0.196 − 0.0813i)11-s + (−0.654 − 1.58i)12-s + 3.95i·13-s + (3.00 − 1.24i)14-s + (−3.18 − 3.18i)15-s + 4.84·16-s + (3.83 + 1.50i)17-s + ⋯
L(s)  = 1  + (−0.807 − 0.807i)2-s + (−1.50 + 0.623i)3-s + 0.303i·4-s + (0.273 + 0.660i)5-s + (1.71 + 0.711i)6-s + (−0.291 + 0.704i)7-s + (−0.562 + 0.562i)8-s + (1.16 − 1.16i)9-s + (0.312 − 0.753i)10-s + (−0.0592 − 0.0245i)11-s + (−0.188 − 0.456i)12-s + 1.09i·13-s + (0.803 − 0.332i)14-s + (−0.822 − 0.822i)15-s + 1.21·16-s + (0.931 + 0.364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.913 - 0.407i$
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: \((2,\ 731,\ (\ :1/2),\ -0.913 - 0.407i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0403749 + 0.189624i\)
\(L(\frac12)\) \(\approx\) \(0.0403749 + 0.189624i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.83 - 1.50i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1.14 + 1.14i)T + 2iT^{2} \)
3 \( 1 + (2.60 - 1.07i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.611 - 1.47i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.771 - 1.86i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.196 + 0.0813i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 3.95iT - 13T^{2} \)
19 \( 1 + (-1.18 - 1.18i)T + 19iT^{2} \)
23 \( 1 + (3.83 + 1.58i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.633 + 1.52i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (3.53 - 1.46i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (8.71 - 3.60i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.83 + 6.85i)T + (-28.9 - 28.9i)T^{2} \)
47 \( 1 - 9.34iT - 47T^{2} \)
53 \( 1 + (2.72 + 2.72i)T + 53iT^{2} \)
59 \( 1 + (2.66 - 2.66i)T - 59iT^{2} \)
61 \( 1 + (5.30 - 12.8i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 0.714T + 67T^{2} \)
71 \( 1 + (12.4 - 5.16i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-0.207 - 0.501i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (2.93 + 1.21i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (10.8 + 10.8i)T + 83iT^{2} \)
89 \( 1 + 16.0iT - 89T^{2} \)
97 \( 1 + (2.75 + 6.64i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.54298937371737749204239007180, −10.22283250622666726875350656724, −9.439147185440347554197426064975, −8.615505151206325252687630469324, −7.13349525010817005603814772075, −5.97537465171537227581311194151, −5.73979329072541846969166686295, −4.40302253436264572569976472069, −3.01027758691135008653439101524, −1.62538606568324659872348087622, 0.18429269989934612789544449368, 1.20129121816076178794003604284, 3.50951622304994076312377160284, 5.14055748931727387784592028509, 5.67983969867881016292313861572, 6.62223323885352667419698499832, 7.38280757258586793296253989199, 7.955263622464730012518206660612, 9.135737994674283321688963165060, 10.02251867127606291371580721157

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