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Properties

Label 2-726-33.29-c1-0-19
Degree $2$
Conductor $726$
Sign $0.578 - 0.815i$
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  + (0.809 + 0.587i)2-s + (−1.65 + 0.514i)3-s + (0.309 + 0.951i)4-s + (0.831 + 1.14i)5-s + (−1.64 − 0.556i)6-s + (4.03 − 1.31i)7-s + (−0.309 + 0.951i)8-s + (2.47 − 1.70i)9-s + 1.41i·10-s + (−1 − 1.41i)12-s + (2.49 − 3.43i)13-s + (4.03 + 1.31i)14-s + (−1.96 − 1.46i)15-s + (−0.809 + 0.587i)16-s + (2.99 + 0.0770i)18-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.954 + 0.296i)3-s + (0.154 + 0.475i)4-s + (0.371 + 0.511i)5-s + (−0.669 − 0.227i)6-s + (1.52 − 0.495i)7-s + (−0.109 + 0.336i)8-s + (0.823 − 0.566i)9-s + 0.447i·10-s + (−0.288 − 0.408i)12-s + (0.691 − 0.951i)13-s + (1.07 + 0.350i)14-s + (−0.506 − 0.378i)15-s + (−0.202 + 0.146i)16-s + (0.706 + 0.0181i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ 0.578 - 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77962 + 0.919052i\)
\(L(\frac12)\) \(\approx\) \(1.77962 + 0.919052i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (1.65 - 0.514i)T \)
11 \( 1 \)
good5 \( 1 + (-0.831 - 1.14i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-4.03 + 1.31i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.49 + 3.43i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.41iT - 23T^{2} \)
29 \( 1 + (-1.85 - 5.70i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.23 - 2.35i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.618 - 1.90i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.85 - 5.70i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.48iT - 43T^{2} \)
47 \( 1 + (9.41 + 3.05i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.15 + 5.72i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (10.7 - 3.49i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.49 - 3.43i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (4.15 + 5.72i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.49 - 3.43i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.70 + 7.05i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 + (6.47 + 4.70i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.69242365897194125305907229629, −10.05703427660864650581451351562, −8.562135644905955213008786044659, −7.80250841517783062540080083069, −6.77859667824464156789057457503, −6.04179415814720457533846824128, −5.04202824584314898350056604507, −4.48674843145361253599518791203, −3.16811680070821641828285700681, −1.36767913838784326792946203070, 1.30359000900874366050399778144, 2.10975783457038230183857160864, 4.10608792316408735246493287198, 4.90735203643186144688813690679, 5.58337553817576345592050956372, 6.43103611652029513893575167741, 7.58869605979109634312452980348, 8.585361258798158788789354995788, 9.528849192795436628382497528331, 10.64502219031840719273655478978

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