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Properties

Label 2-726-33.32-c1-0-16
Degree $2$
Conductor $726$
Sign $0.244 + 0.969i$
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 + (−1.71 − 0.232i)3-s + 4-s − 2.65i·5-s + (1.71 + 0.232i)6-s + 1.64i·7-s − 8-s + (2.89 + 0.798i)9-s + 2.65i·10-s + (−1.71 − 0.232i)12-s − 1.50i·13-s − 1.64i·14-s + (−0.618 + 4.55i)15-s + 16-s + 6.79·17-s + (−2.89 − 0.798i)18-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.990 − 0.134i)3-s + 0.5·4-s − 1.18i·5-s + (0.700 + 0.0950i)6-s + 0.620i·7-s − 0.353·8-s + (0.963 + 0.266i)9-s + 0.839i·10-s + (−0.495 − 0.0671i)12-s − 0.417i·13-s − 0.438i·14-s + (−0.159 + 1.17i)15-s + 0.250·16-s + 1.64·17-s + (−0.681 − 0.188i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $0.244 + 0.969i$
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: \((2,\ 726,\ (\ :1/2),\ 0.244 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.583415 - 0.454513i\)
\(L(\frac12)\) \(\approx\) \(0.583415 - 0.454513i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.232i)T \)
11 \( 1 \)
good5 \( 1 + 2.65iT - 5T^{2} \)
7 \( 1 - 1.64iT - 7T^{2} \)
13 \( 1 + 1.50iT - 13T^{2} \)
17 \( 1 - 6.79T + 17T^{2} \)
19 \( 1 - 5.48iT - 19T^{2} \)
23 \( 1 + 4.78iT - 23T^{2} \)
29 \( 1 + 0.464T + 29T^{2} \)
31 \( 1 + 5.19T + 31T^{2} \)
37 \( 1 - 1.00T + 37T^{2} \)
41 \( 1 - 1.57T + 41T^{2} \)
43 \( 1 + 3.42iT - 43T^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 + 9.82iT - 53T^{2} \)
59 \( 1 + 6.11iT - 59T^{2} \)
61 \( 1 + 13.9iT - 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 - 0.898iT - 71T^{2} \)
73 \( 1 + 4.62iT - 73T^{2} \)
79 \( 1 - 3.17iT - 79T^{2} \)
83 \( 1 - 1.87T + 83T^{2} \)
89 \( 1 - 12.8iT - 89T^{2} \)
97 \( 1 + 3.59T + 97T^{2} \)
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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

−10.12012918504139245328804712742, −9.494719194531819219425506509434, −8.398154586871839143677954069867, −7.86151443713637298739991969885, −6.67147061462218070304059939256, −5.56872585142622238152157650938, −5.22387373081193829433583255211, −3.73187501130226090841422127020, −1.86159390927664346840555308477, −0.66502705697836599546299529457, 1.17488008535317504335501177556, 2.90715480048940400665121668048, 4.06237380214512920030375751957, 5.44093711386824693266359879028, 6.32230147060021816062500037964, 7.29260077942374622805498086898, 7.50951760833286748197403808372, 9.164351074386196161851131151699, 9.916096095230029023668246695048, 10.62176624106013156607041242869

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