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Properties

Label 2-728-13.12-c1-0-10
Degree $2$
Conductor $728$
Sign $0.680 - 0.732i$
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  + 2.45·3-s + 1.45i·5-s + i·7-s + 3.02·9-s + 1.64i·11-s + (−2.45 + 2.64i)13-s + 3.57i·15-s + 5.29·17-s + 0.640i·19-s + 2.45i·21-s + 2.18·23-s + 2.88·25-s + 0.0679·27-s − 2.21·29-s − 4.29i·31-s + ⋯
L(s)  = 1  + 1.41·3-s + 0.650i·5-s + 0.377i·7-s + 1.00·9-s + 0.494i·11-s + (−0.680 + 0.732i)13-s + 0.922i·15-s + 1.28·17-s + 0.146i·19-s + 0.535i·21-s + 0.455·23-s + 0.576·25-s + 0.0130·27-s − 0.410·29-s − 0.771i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.680 - 0.732i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.680 - 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18997 + 0.954124i\)
\(L(\frac12)\) \(\approx\) \(2.18997 + 0.954124i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - iT \)
13 \( 1 + (2.45 - 2.64i)T \)
good3 \( 1 - 2.45T + 3T^{2} \)
5 \( 1 - 1.45iT - 5T^{2} \)
11 \( 1 - 1.64iT - 11T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 - 0.640iT - 19T^{2} \)
23 \( 1 - 2.18T + 23T^{2} \)
29 \( 1 + 2.21T + 29T^{2} \)
31 \( 1 + 4.29iT - 31T^{2} \)
37 \( 1 + 3.26iT - 37T^{2} \)
41 \( 1 + 0.842iT - 41T^{2} \)
43 \( 1 - 5.12T + 43T^{2} \)
47 \( 1 + 0.572iT - 47T^{2} \)
53 \( 1 + 6.25T + 53T^{2} \)
59 \( 1 + 10.0iT - 59T^{2} \)
61 \( 1 + 7.21T + 61T^{2} \)
67 \( 1 + 3.75iT - 67T^{2} \)
71 \( 1 + 2.42iT - 71T^{2} \)
73 \( 1 - 1.48iT - 73T^{2} \)
79 \( 1 - 5.19T + 79T^{2} \)
83 \( 1 + 17.3iT - 83T^{2} \)
89 \( 1 + 4.69iT - 89T^{2} \)
97 \( 1 - 11.5iT - 97T^{2} \)
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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

−10.24910013283834941416395558757, −9.478559202493289456757879809477, −8.943682688678308190416181744332, −7.78046899574427402019422539696, −7.37925190721741003966193763093, −6.23958754729392613780719755622, −4.93660083333094557335005978886, −3.71372411092747995266767774589, −2.84285531148662330179914853537, −1.93058100726791730632800016855, 1.17755346752735106484043221106, 2.75874112674949245185134497027, 3.48179348552622821552352367398, 4.70211730684332933812878908665, 5.68770874595232428069666533383, 7.14817257108469165409561858862, 7.87485860474854071973263832987, 8.534570644651744537546515355709, 9.306917959490403585233162825549, 10.03777840470360461808345542179

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