Properties

Label 2-736-736.229-c0-0-2
Degree $2$
Conductor $736$
Sign $-0.555 + 0.831i$
Analytic cond. $0.367311$
Root an. cond. $0.606062$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.292 − 0.707i)3-s − 1.00i·4-s + (−0.707 − 0.292i)6-s + (−0.707 − 0.707i)8-s + (0.292 − 0.292i)9-s + (−0.707 + 0.292i)12-s + (−0.707 + 0.292i)13-s − 1.00·16-s − 0.414i·18-s + (0.707 − 0.707i)23-s + (−0.292 + 0.707i)24-s + (0.707 + 0.707i)25-s + (−0.292 + 0.707i)26-s + (−1.00 − 0.414i)27-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.292 − 0.707i)3-s − 1.00i·4-s + (−0.707 − 0.292i)6-s + (−0.707 − 0.707i)8-s + (0.292 − 0.292i)9-s + (−0.707 + 0.292i)12-s + (−0.707 + 0.292i)13-s − 1.00·16-s − 0.414i·18-s + (0.707 − 0.707i)23-s + (−0.292 + 0.707i)24-s + (0.707 + 0.707i)25-s + (−0.292 + 0.707i)26-s + (−1.00 − 0.414i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $-0.555 + 0.831i$
Analytic conductor: \(0.367311\)
Root analytic conductor: \(0.606062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{736} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 736,\ (\ :0),\ -0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.212998940\)
\(L(\frac12)\) \(\approx\) \(1.212998940\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.707 - 0.707i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.707 - 0.707i)T^{2} \)
67 \( 1 + (0.707 - 0.707i)T^{2} \)
71 \( 1 + (-1 - i)T + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 - T^{2} \)
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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

−10.62485614334049302764636418820, −9.521699930444761900108599395962, −8.854798224601706717704215931906, −7.24810942386533926037956600035, −6.82928280521581485555175099292, −5.70192004126988417147934157151, −4.83451135669370242503742642095, −3.72210337633755970751181008784, −2.50391980715429309966556750316, −1.22540104101849949540444023662, 2.51411695983319555965255557387, 3.78732045952663410592269277917, 4.69294199032894607211514124687, 5.34219861925790361196128871375, 6.35914796790331778082418912681, 7.37511964800938495720610200980, 8.055088308902304451384487995997, 9.209051860820327037382941814477, 9.979628900338817588110806712111, 10.98090532195431937098809594065

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