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Properties

Label 2-928-29.19-c0-0-0
Degree $2$
Conductor $928$
Sign $0.891 + 0.452i$
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  + (−1.52 − 0.347i)5-s + (0.781 + 0.623i)9-s + (1.40 − 1.12i)13-s + (0.752 − 0.752i)17-s + (1.30 + 0.626i)25-s + (0.781 − 0.623i)29-s + (0.656 − 0.0739i)37-s + (−1.33 − 1.33i)41-s + (−0.974 − 1.22i)45-s + (−0.623 + 0.781i)49-s + (−0.433 + 1.90i)53-s + (−0.559 + 1.59i)61-s + (−2.53 + 1.22i)65-s + (0.900 − 1.43i)73-s + (0.222 + 0.974i)81-s + ⋯
L(s)  = 1  + (−1.52 − 0.347i)5-s + (0.781 + 0.623i)9-s + (1.40 − 1.12i)13-s + (0.752 − 0.752i)17-s + (1.30 + 0.626i)25-s + (0.781 − 0.623i)29-s + (0.656 − 0.0739i)37-s + (−1.33 − 1.33i)41-s + (−0.974 − 1.22i)45-s + (−0.623 + 0.781i)49-s + (−0.433 + 1.90i)53-s + (−0.559 + 1.59i)61-s + (−2.53 + 1.22i)65-s + (0.900 − 1.43i)73-s + (0.222 + 0.974i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(928\)    =    \(2^{5} \cdot 29\)
Sign: $0.891 + 0.452i$
Analytic conductor: \(0.463132\)
Root analytic conductor: \(0.680538\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{928} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 928,\ (\ :0),\ 0.891 + 0.452i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8686984074\)
\(L(\frac12)\) \(\approx\) \(0.8686984074\)
\(L(1)\) 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 \( 1 + (-0.781 + 0.623i)T \)
good3 \( 1 + (-0.781 - 0.623i)T^{2} \)
5 \( 1 + (1.52 + 0.347i)T + (0.900 + 0.433i)T^{2} \)
7 \( 1 + (0.623 - 0.781i)T^{2} \)
11 \( 1 + (0.974 + 0.222i)T^{2} \)
13 \( 1 + (-1.40 + 1.12i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.752 + 0.752i)T - iT^{2} \)
19 \( 1 + (0.781 - 0.623i)T^{2} \)
23 \( 1 + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.433 + 0.900i)T^{2} \)
37 \( 1 + (-0.656 + 0.0739i)T + (0.974 - 0.222i)T^{2} \)
41 \( 1 + (1.33 + 1.33i)T + iT^{2} \)
43 \( 1 + (0.433 + 0.900i)T^{2} \)
47 \( 1 + (-0.974 - 0.222i)T^{2} \)
53 \( 1 + (0.433 - 1.90i)T + (-0.900 - 0.433i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (0.559 - 1.59i)T + (-0.781 - 0.623i)T^{2} \)
67 \( 1 + (0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.900 + 1.43i)T + (-0.433 - 0.900i)T^{2} \)
79 \( 1 + (-0.974 + 0.222i)T^{2} \)
83 \( 1 + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-1.05 - 1.68i)T + (-0.433 + 0.900i)T^{2} \)
97 \( 1 + (0.218 + 0.623i)T + (-0.781 + 0.623i)T^{2} \)
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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

−10.45618697103013040441758001976, −9.294384628197970131926632854216, −8.223381306396833874304318605908, −7.88277257263276987749104994936, −7.06055491907160490930066893350, −5.80813822380457488850619699437, −4.76444036464044658159407292397, −3.94427765395665784011755561435, −3.02559197000787974559010211312, −1.07738517734723118957392042135, 1.40936106990162879311051470072, 3.41997989777306363497619921353, 3.83332625637441541170774250405, 4.81885129534054491186322696612, 6.42761666025390810446778993176, 6.79553659978509375871066959073, 7.966457489255048925605729405237, 8.434853953891908997099269294262, 9.505775998147763574491798699712, 10.42195329414409723608940135568

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