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Properties

Label 2-728-56.27-c1-0-53
Degree $2$
Conductor $728$
Sign $0.995 + 0.0908i$
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  + (−1.10 + 0.887i)2-s + 0.140i·3-s + (0.424 − 1.95i)4-s + 0.986·5-s + (−0.124 − 0.154i)6-s + (1.39 − 2.24i)7-s + (1.26 + 2.52i)8-s + 2.98·9-s + (−1.08 + 0.875i)10-s − 1.55·11-s + (0.273 + 0.0595i)12-s + 13-s + (0.459 + 3.71i)14-s + 0.138i·15-s + (−3.63 − 1.66i)16-s + 4.36i·17-s + ⋯
L(s)  = 1  + (−0.778 + 0.627i)2-s + 0.0809i·3-s + (0.212 − 0.977i)4-s + 0.441·5-s + (−0.0507 − 0.0630i)6-s + (0.527 − 0.849i)7-s + (0.447 + 0.894i)8-s + 0.993·9-s + (−0.343 + 0.276i)10-s − 0.469·11-s + (0.0790 + 0.0171i)12-s + 0.277·13-s + (0.122 + 0.992i)14-s + 0.0357i·15-s + (−0.909 − 0.415i)16-s + 1.05i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.995 + 0.0908i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.995 + 0.0908i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24027 - 0.0564528i\)
\(L(\frac12)\) \(\approx\) \(1.24027 - 0.0564528i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 - 0.887i)T \)
7 \( 1 + (-1.39 + 2.24i)T \)
13 \( 1 - T \)
good3 \( 1 - 0.140iT - 3T^{2} \)
5 \( 1 - 0.986T + 5T^{2} \)
11 \( 1 + 1.55T + 11T^{2} \)
17 \( 1 - 4.36iT - 17T^{2} \)
19 \( 1 + 7.69iT - 19T^{2} \)
23 \( 1 + 2.65iT - 23T^{2} \)
29 \( 1 + 5.33iT - 29T^{2} \)
31 \( 1 - 9.27T + 31T^{2} \)
37 \( 1 - 0.0671iT - 37T^{2} \)
41 \( 1 - 5.51iT - 41T^{2} \)
43 \( 1 + 5.43T + 43T^{2} \)
47 \( 1 - 6.75T + 47T^{2} \)
53 \( 1 + 3.62iT - 53T^{2} \)
59 \( 1 - 4.10iT - 59T^{2} \)
61 \( 1 - 8.11T + 61T^{2} \)
67 \( 1 - 5.12T + 67T^{2} \)
71 \( 1 - 1.96iT - 71T^{2} \)
73 \( 1 - 3.52iT - 73T^{2} \)
79 \( 1 + 9.12iT - 79T^{2} \)
83 \( 1 - 5.07iT - 83T^{2} \)
89 \( 1 - 1.10iT - 89T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−10.21300586945396416977634813722, −9.661316540217012933074022803625, −8.511610530134139379920542202446, −7.85313344798765492061244068843, −6.93258684869463681723060141230, −6.24519669543334160886750153564, −4.99383737780306192443614744977, −4.21303965682666126476598571552, −2.25772843672368228822759666935, −0.960646908982586775750497284472, 1.40815057736247320862987367094, 2.36206708161797591877464755395, 3.65236818659120883402013284053, 4.91335901599997854809461986552, 6.02332275039422254080743679820, 7.20934190168353736019617068996, 7.982885405792476974353935526514, 8.761153041055879518195088102629, 9.776279027039050551644114999881, 10.14919049511641900189886831067

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