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Properties

Label 1-920-920.613-r0-0-0
Degree $1$
Conductor $920$
Sign $-0.907 - 0.418i$
Analytic cond. $4.27246$
Root an. cond. $4.27246$
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  + (0.755 + 0.654i)3-s + (−0.909 + 0.415i)7-s + (0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (−0.909 − 0.415i)13-s + (0.540 + 0.841i)17-s + (−0.841 − 0.540i)19-s + (−0.959 − 0.281i)21-s + (−0.540 + 0.841i)27-s + (0.841 − 0.540i)29-s + (−0.654 − 0.755i)31-s + (−0.909 − 0.415i)33-s + (−0.989 + 0.142i)37-s + (−0.415 − 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)3-s + (−0.909 + 0.415i)7-s + (0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (−0.909 − 0.415i)13-s + (0.540 + 0.841i)17-s + (−0.841 − 0.540i)19-s + (−0.959 − 0.281i)21-s + (−0.540 + 0.841i)27-s + (0.841 − 0.540i)29-s + (−0.654 − 0.755i)31-s + (−0.909 − 0.415i)33-s + (−0.989 + 0.142i)37-s + (−0.415 − 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.907 - 0.418i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ -0.907 - 0.418i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07093418578 + 0.3230306727i\)
\(L(\frac12)\) \(\approx\) \(-0.07093418578 + 0.3230306727i\)
\(L(1)\) \(\approx\) \(0.8161701172 + 0.2994146722i\)
\(L(1)\) \(\approx\) \(0.8161701172 + 0.2994146722i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.755 + 0.654i)T \)
7 \( 1 + (-0.909 + 0.415i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (-0.909 - 0.415i)T \)
17 \( 1 + (0.540 + 0.841i)T \)
19 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (0.841 - 0.540i)T \)
31 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + (-0.989 + 0.142i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.909 + 0.415i)T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (-0.654 - 0.755i)T \)
67 \( 1 + (-0.281 + 0.959i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (-0.540 + 0.841i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (0.989 - 0.142i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (-0.989 - 0.142i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.19917316183236240187372104323, −20.64802731576505191348549740218, −19.61365442161040570607022665320, −19.232895800485267275552428032297, −18.44951986929785875150339076008, −17.62523080816789861028618015709, −16.55778857923696464084491374223, −15.93119132762447214028287563640, −14.90045824461162518754923655130, −14.09446396766656242442416205194, −13.492697923016461949754502777966, −12.54070457585185033087532119437, −12.1685212770708809490716060748, −10.71568180523747069980348585515, −9.944110616806968949179374031743, −9.137869356200991965363848468497, −8.22589984237684453394244270152, −7.314552607003128861469348233925, −6.78335874085749667967058111015, −5.689300575009091728155249130982, −4.5142002265970751066737488332, −3.30150706190089175366436295752, −2.74868448348458153426065633704, −1.58816268319017730805223826451, −0.11789519954353322643899546048, 2.05884215841706994482317739721, 2.78562988823454313866415537208, 3.62697914524166298318098100753, 4.72567715792879767770563959284, 5.51425172891804148773678619617, 6.657789814512237791343641464499, 7.73862296824238223369665637650, 8.42904791022094843815064229548, 9.38547014664572080273849265519, 10.13556041568715057797115047379, 10.60232410977524258339475625259, 11.99448237257381569439516417017, 12.90284946171438637441010275565, 13.38031728370089822196805100542, 14.6062379583244134313305944612, 15.20104358729669180013171302182, 15.76462587178980832480187728730, 16.67699925524783278951157769272, 17.45516943908891392353319715427, 18.7000317521640783759129452163, 19.238408857618370867962331754063, 19.96731049593131789140979101043, 20.729536378208434382278544589200, 21.68684362760669540944451185303, 21.99340394115557264428373795792

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