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Properties

Label 2-728-728.461-c1-0-4
Degree $2$
Conductor $728$
Sign $-0.860 - 0.508i$
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  + (−0.366 − 1.36i)2-s + (−0.636 + 1.10i)3-s + (−1.73 + i)4-s + (−0.465 + 0.465i)5-s + (1.73 + 0.465i)6-s + (−0.684 + 2.55i)7-s + (2 + 1.99i)8-s + (0.689 + 1.19i)9-s + (0.807 + 0.465i)10-s − 2.54i·12-s + (−0.102 − 3.60i)13-s + 3.74·14-s + (−0.217 − 0.810i)15-s + (1.99 − 3.46i)16-s + (1.37 − 1.37i)18-s + (−7.36 − 1.97i)19-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.367 + 0.636i)3-s + (−0.866 + 0.5i)4-s + (−0.208 + 0.208i)5-s + (0.709 + 0.190i)6-s + (−0.258 + 0.965i)7-s + (0.707 + 0.707i)8-s + (0.229 + 0.398i)9-s + (0.255 + 0.147i)10-s − 0.735i·12-s + (−0.0284 − 0.999i)13-s + 0.999·14-s + (−0.0560 − 0.209i)15-s + (0.499 − 0.866i)16-s + (0.325 − 0.325i)18-s + (−1.69 − 0.452i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.860 - 0.508i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ -0.860 - 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0652441 + 0.238624i\)
\(L(\frac12)\) \(\approx\) \(0.0652441 + 0.238624i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
7 \( 1 + (0.684 - 2.55i)T \)
13 \( 1 + (0.102 + 3.60i)T \)
good3 \( 1 + (0.636 - 1.10i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.465 - 0.465i)T - 5iT^{2} \)
11 \( 1 + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.36 + 1.97i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.01 + 1.74i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 31iT^{2} \)
37 \( 1 + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (3.35 - 12.5i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (7.77 + 13.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (16.0 + 4.29i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 - 8.25T + 79T^{2} \)
83 \( 1 + (4.00 - 4.00i)T - 83iT^{2} \)
89 \( 1 + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-84.0 - 48.5i)T^{2} \)
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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

−10.70712324460709729705977220793, −10.12713933662446218119771251781, −9.214931926902479206614245226792, −8.446301769361946638590173723833, −7.56517274192053396122705369945, −6.09063467451888288714503670939, −5.12104705048397543861131495207, −4.25420568744673253008654157702, −3.11311032912301199113347918327, −2.05085024351402051369105920173, 0.14668513627062788227937934034, 1.59435055305456697360215051448, 3.93845226643998305230010784173, 4.49346936447538378198986905397, 6.02616119867159604719137113636, 6.55183322714067573959079014489, 7.28908716189033445666212025116, 8.112224155332837948101903068761, 9.032654152181544780501766919342, 9.942089036439574040466107846365

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