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Properties

Label 2-928-29.8-c0-0-0
Degree $2$
Conductor $928$
Sign $0.505 - 0.863i$
Analytic cond. $0.463132$
Root an. cond. $0.680538$
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  + (0.846 + 1.75i)5-s + (0.974 + 0.222i)9-s + (−1.21 + 0.277i)13-s + (−1.33 − 1.33i)17-s + (−1.74 + 2.19i)25-s + (0.974 − 0.222i)29-s + (0.119 + 0.189i)37-s + (1.40 − 1.40i)41-s + (0.433 + 1.90i)45-s + (0.222 − 0.974i)49-s + (0.781 − 0.376i)53-s + (−0.0739 + 0.656i)61-s + (−1.51 − 1.90i)65-s + (−0.623 + 0.218i)73-s + (0.900 + 0.433i)81-s + ⋯
L(s)  = 1  + (0.846 + 1.75i)5-s + (0.974 + 0.222i)9-s + (−1.21 + 0.277i)13-s + (−1.33 − 1.33i)17-s + (−1.74 + 2.19i)25-s + (0.974 − 0.222i)29-s + (0.119 + 0.189i)37-s + (1.40 − 1.40i)41-s + (0.433 + 1.90i)45-s + (0.222 − 0.974i)49-s + (0.781 − 0.376i)53-s + (−0.0739 + 0.656i)61-s + (−1.51 − 1.90i)65-s + (−0.623 + 0.218i)73-s + (0.900 + 0.433i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(928\)    =    \(2^{5} \cdot 29\)
Sign: $0.505 - 0.863i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{928} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 928,\ (\ :0),\ 0.505 - 0.863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.137248754\)
\(L(\frac12)\) \(\approx\) \(1.137248754\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (-0.974 + 0.222i)T \)
good3 \( 1 + (-0.974 - 0.222i)T^{2} \)
5 \( 1 + (-0.846 - 1.75i)T + (-0.623 + 0.781i)T^{2} \)
7 \( 1 + (-0.222 + 0.974i)T^{2} \)
11 \( 1 + (-0.433 - 0.900i)T^{2} \)
13 \( 1 + (1.21 - 0.277i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (1.33 + 1.33i)T + iT^{2} \)
19 \( 1 + (0.974 - 0.222i)T^{2} \)
23 \( 1 + (0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.781 + 0.623i)T^{2} \)
37 \( 1 + (-0.119 - 0.189i)T + (-0.433 + 0.900i)T^{2} \)
41 \( 1 + (-1.40 + 1.40i)T - iT^{2} \)
43 \( 1 + (-0.781 + 0.623i)T^{2} \)
47 \( 1 + (0.433 + 0.900i)T^{2} \)
53 \( 1 + (-0.781 + 0.376i)T + (0.623 - 0.781i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (0.0739 - 0.656i)T + (-0.974 - 0.222i)T^{2} \)
67 \( 1 + (0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.623 - 0.218i)T + (0.781 - 0.623i)T^{2} \)
79 \( 1 + (0.433 - 0.900i)T^{2} \)
83 \( 1 + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (1.00 + 0.351i)T + (0.781 + 0.623i)T^{2} \)
97 \( 1 + (0.0250 + 0.222i)T + (-0.974 + 0.222i)T^{2} \)
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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

−10.25002256140600901160603702885, −9.846606237144278539677200287042, −8.994071176713151590743308624577, −7.38672814547944981297476640792, −7.11484277346349985780136569316, −6.36937779194796771014282934971, −5.20249142941940536568938174480, −4.15089716902231224728286100881, −2.72077514741668193095369283259, −2.17322320659880608553443512972, 1.25688612028746739677871922853, 2.33767266693355618366744813782, 4.29556412828544791561007489774, 4.66297632067182976555211084246, 5.74106004265636997114465512248, 6.58683651559968779992289495162, 7.77961260192248744421823233587, 8.594233867376151455188199567999, 9.357425342485288342588557002032, 9.910389816653575359381255474608

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