Properties

Label 2-736-736.229-c0-0-1
Degree $2$
Conductor $736$
Sign $0.997 + 0.0654i$
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  + (0.258 + 0.965i)2-s + (−0.465 − 1.12i)3-s + (−0.866 + 0.499i)4-s + (0.965 − 0.741i)6-s + (−0.707 − 0.707i)8-s + (−0.341 + 0.341i)9-s + (0.965 + 0.741i)12-s + (1.83 − 0.758i)13-s + (0.500 − 0.866i)16-s + (−0.417 − 0.241i)18-s + (0.707 − 0.707i)23-s + (−0.465 + 1.12i)24-s + (0.707 + 0.707i)25-s + (1.20 + 1.57i)26-s + (−0.582 − 0.241i)27-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.465 − 1.12i)3-s + (−0.866 + 0.499i)4-s + (0.965 − 0.741i)6-s + (−0.707 − 0.707i)8-s + (−0.341 + 0.341i)9-s + (0.965 + 0.741i)12-s + (1.83 − 0.758i)13-s + (0.500 − 0.866i)16-s + (−0.417 − 0.241i)18-s + (0.707 − 0.707i)23-s + (−0.465 + 1.12i)24-s + (0.707 + 0.707i)25-s + (1.20 + 1.57i)26-s + (−0.582 − 0.241i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $0.997 + 0.0654i$
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: \((2,\ 736,\ (\ :0),\ 0.997 + 0.0654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8957220267\)
\(L(\frac12)\) \(\approx\) \(0.8957220267\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (0.465 + 1.12i)T + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.707 - 0.707i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (-1.83 + 0.758i)T + (0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.607 + 1.46i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + 0.517T + T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 - 1.73iT - T^{2} \)
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 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
73 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
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

−10.74622061500521607728691598813, −9.438467854550706007537757129586, −8.529200978785741882338364284784, −7.80801610981498567619856285449, −6.99612751510290031955882974491, −6.15256888597325410198076123983, −5.68451931182306142016678900766, −4.34630226494729509342794289696, −3.14806699531608197412948159583, −1.14162771665632610666043139144, 1.62876957568802479764486498609, 3.39314287335250416122545151853, 3.98006364950520574354074020278, 5.02073249293762423000204988920, 5.71837000561249653974221080361, 6.92009066084390567797991074219, 8.651347145968026258725935277170, 8.990521396619049448942164780405, 10.04983836210746837282589077067, 10.69352661872837659289274477725

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