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Properties

Label 2-728-56.27-c1-0-74
Degree $2$
Conductor $728$
Sign $-0.993 + 0.113i$
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  + (−1.39 − 0.227i)2-s − 1.35i·3-s + (1.89 + 0.635i)4-s − 2.01·5-s + (−0.308 + 1.89i)6-s + (2.18 − 1.49i)7-s + (−2.50 − 1.31i)8-s + 1.16·9-s + (2.81 + 0.458i)10-s − 4.92·11-s + (0.861 − 2.57i)12-s + 13-s + (−3.38 + 1.58i)14-s + 2.73i·15-s + (3.19 + 2.40i)16-s − 4.50i·17-s + ⋯
L(s)  = 1  + (−0.986 − 0.160i)2-s − 0.783i·3-s + (0.948 + 0.317i)4-s − 0.902·5-s + (−0.125 + 0.772i)6-s + (0.825 − 0.563i)7-s + (−0.884 − 0.465i)8-s + 0.386·9-s + (0.890 + 0.145i)10-s − 1.48·11-s + (0.248 − 0.742i)12-s + 0.277·13-s + (−0.905 + 0.423i)14-s + 0.706i·15-s + (0.798 + 0.602i)16-s − 1.09i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.993 + 0.113i$
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: \((2,\ 728,\ (\ :1/2),\ -0.993 + 0.113i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0284796 - 0.498299i\)
\(L(\frac12)\) \(\approx\) \(0.0284796 - 0.498299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.227i)T \)
7 \( 1 + (-2.18 + 1.49i)T \)
13 \( 1 - T \)
good3 \( 1 + 1.35iT - 3T^{2} \)
5 \( 1 + 2.01T + 5T^{2} \)
11 \( 1 + 4.92T + 11T^{2} \)
17 \( 1 + 4.50iT - 17T^{2} \)
19 \( 1 + 4.26iT - 19T^{2} \)
23 \( 1 - 9.28iT - 23T^{2} \)
29 \( 1 + 6.03iT - 29T^{2} \)
31 \( 1 + 2.67T + 31T^{2} \)
37 \( 1 - 1.91iT - 37T^{2} \)
41 \( 1 + 9.84iT - 41T^{2} \)
43 \( 1 + 1.98T + 43T^{2} \)
47 \( 1 + 6.61T + 47T^{2} \)
53 \( 1 - 7.90iT - 53T^{2} \)
59 \( 1 - 9.77iT - 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + 15.8T + 67T^{2} \)
71 \( 1 - 2.17iT - 71T^{2} \)
73 \( 1 - 7.87iT - 73T^{2} \)
79 \( 1 + 6.79iT - 79T^{2} \)
83 \( 1 + 15.5iT - 83T^{2} \)
89 \( 1 + 5.33iT - 89T^{2} \)
97 \( 1 - 5.40iT - 97T^{2} \)
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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

−10.05610496570334576420210931176, −9.011780923688937825313353580360, −7.903929192716763075407409654984, −7.54915382756381218996124997889, −7.12728322046436340411715721402, −5.63620427624988253548338137707, −4.36000048247686737286947230699, −2.99224809967486866202696192118, −1.71165811173754747403139438039, −0.35051623320574303605017517357, 1.78879211337411334562441706347, 3.22066383026715173980426037439, 4.49402356974667769108442038946, 5.41094598408980852751164205708, 6.55513924700156091832343224030, 7.955365942039305729138940799611, 8.027582019057267988943071330822, 8.978366636383447135070624087780, 10.12109627150974942322234034523, 10.65109696320304707746709653744

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