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Properties

Label 2-726-33.32-c1-0-20
Degree $2$
Conductor $726$
Sign $0.682 - 0.731i$
Analytic cond. $5.79713$
Root an. cond. $2.40772$
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-s + (1.69 + 0.346i)3-s + 4-s + 3.90i·5-s + (1.69 + 0.346i)6-s − 3.17i·7-s + 8-s + (2.76 + 1.17i)9-s + 3.90i·10-s + (1.69 + 0.346i)12-s + 2.17i·13-s − 3.17i·14-s + (−1.35 + 6.62i)15-s + 16-s − 2.64·17-s + (2.76 + 1.17i)18-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.979 + 0.199i)3-s + 0.5·4-s + 1.74i·5-s + (0.692 + 0.141i)6-s − 1.19i·7-s + 0.353·8-s + (0.920 + 0.391i)9-s + 1.23i·10-s + (0.489 + 0.0999i)12-s + 0.603i·13-s − 0.847i·14-s + (−0.348 + 1.71i)15-s + 0.250·16-s − 0.640·17-s + (0.650 + 0.277i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $0.682 - 0.731i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (725, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ 0.682 - 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.93849 + 1.27725i\)
\(L(\frac12)\) \(\approx\) \(2.93849 + 1.27725i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.69 - 0.346i)T \)
11 \( 1 \)
good5 \( 1 - 3.90iT - 5T^{2} \)
7 \( 1 + 3.17iT - 7T^{2} \)
13 \( 1 - 2.17iT - 13T^{2} \)
17 \( 1 + 2.64T + 17T^{2} \)
19 \( 1 + 0.214iT - 19T^{2} \)
23 \( 1 + 4.93iT - 23T^{2} \)
29 \( 1 - 7.54T + 29T^{2} \)
31 \( 1 + 1.14T + 31T^{2} \)
37 \( 1 + 2.87T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 + 1.44iT - 43T^{2} \)
47 \( 1 + 4.76iT - 47T^{2} \)
53 \( 1 - 3.44iT - 53T^{2} \)
59 \( 1 + 0.506iT - 59T^{2} \)
61 \( 1 - 2.62iT - 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 + 2.42iT - 73T^{2} \)
79 \( 1 + 14.3iT - 79T^{2} \)
83 \( 1 + 1.33T + 83T^{2} \)
89 \( 1 + 7.45iT - 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−10.41322768062500259616386531594, −10.09316706973924945026154547142, −8.708817046493421019042524323056, −7.63926086914223179882368895776, −6.86535979412415067146677613811, −6.52764980754364894275956294225, −4.69811155157608957417846152927, −3.84323710362140225749534988145, −3.07248076121517531459116250371, −2.07158923871801570059402762740, 1.43358601923576908224260071861, 2.56787731913816329681180127546, 3.79298823974575726004502533802, 4.89025559892211922465220022907, 5.50023647135104486157643654984, 6.69135463805168612380596895370, 8.027253080919108770001271566996, 8.500686685524968104196215284442, 9.216290017803138261577660818139, 10.02038632957506262006493232522

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