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Properties

Label 2-728-104.101-c1-0-34
Degree $2$
Conductor $728$
Sign $-0.277 - 0.960i$
Analytic cond. $5.81310$
Root an. cond. $2.41103$
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  + (0.105 + 1.41i)2-s + (0.380 − 0.219i)3-s + (−1.97 + 0.296i)4-s + 1.63·5-s + (0.349 + 0.513i)6-s + (0.866 + 0.5i)7-s + (−0.625 − 2.75i)8-s + (−1.40 + 2.43i)9-s + (0.171 + 2.30i)10-s + (0.958 + 1.65i)11-s + (−0.687 + 0.547i)12-s + (3.59 − 0.256i)13-s + (−0.614 + 1.27i)14-s + (0.620 − 0.358i)15-s + (3.82 − 1.17i)16-s + (2.70 − 4.68i)17-s + ⋯
L(s)  = 1  + (0.0742 + 0.997i)2-s + (0.219 − 0.126i)3-s + (−0.988 + 0.148i)4-s + 0.729·5-s + (0.142 + 0.209i)6-s + (0.327 + 0.188i)7-s + (−0.221 − 0.975i)8-s + (−0.467 + 0.810i)9-s + (0.0541 + 0.727i)10-s + (0.288 + 0.500i)11-s + (−0.198 + 0.157i)12-s + (0.997 − 0.0711i)13-s + (−0.164 + 0.340i)14-s + (0.160 − 0.0925i)15-s + (0.956 − 0.292i)16-s + (0.655 − 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.277 - 0.960i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ -0.277 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06036 + 1.41025i\)
\(L(\frac12)\) \(\approx\) \(1.06036 + 1.41025i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.105 - 1.41i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-3.59 + 0.256i)T \)
good3 \( 1 + (-0.380 + 0.219i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.63T + 5T^{2} \)
11 \( 1 + (-0.958 - 1.65i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.70 + 4.68i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.50 - 2.61i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.32 - 5.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.30 - 4.79i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.44iT - 31T^{2} \)
37 \( 1 + (-1.88 - 3.26i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.34 + 4.81i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.67 - 5.58i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.314iT - 47T^{2} \)
53 \( 1 - 1.36iT - 53T^{2} \)
59 \( 1 + (6.13 - 10.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.24 + 4.18i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.26 + 2.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (14.4 + 8.33i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 + 1.03T + 79T^{2} \)
83 \( 1 - 6.81T + 83T^{2} \)
89 \( 1 + (-6.37 + 3.67i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.88 - 1.09i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.51626829849645448459845612067, −9.303495620125479733653135530535, −9.083045594993872280579959401165, −7.72870599124237769305226576995, −7.45293839069229785944752306287, −5.96689884809542896898436589187, −5.59803680708167554000592933979, −4.51000500809399939231900006496, −3.19487719636791079604011945348, −1.61072610961187008456816303551, 0.990773779252789091190887081715, 2.31328237743223257518102669065, 3.52179016065166877723016607223, 4.27507596887252682977602512840, 5.74631806658168566831701518394, 6.19244632346348262133754430068, 7.87456989610557211547427925847, 8.869665613396903915556807967706, 9.201418028178339427336872342582, 10.28502159664990562527975876148

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