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Properties

Label 1-40e2-1600.269-r0-0-0
Degree $1$
Conductor $1600$
Sign $0.829 + 0.558i$
Analytic cond. $7.43036$
Root an. cond. $7.43036$
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  + (0.760 + 0.649i)3-s + (0.707 − 0.707i)7-s + (0.156 + 0.987i)9-s + (0.852 + 0.522i)11-s + (0.522 + 0.852i)13-s + (0.951 − 0.309i)17-s + (0.0784 − 0.996i)19-s + (0.996 − 0.0784i)21-s + (0.987 + 0.156i)23-s + (−0.522 + 0.852i)27-s + (0.760 + 0.649i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (−0.972 − 0.233i)37-s + (−0.156 + 0.987i)39-s + ⋯
L(s)  = 1  + (0.760 + 0.649i)3-s + (0.707 − 0.707i)7-s + (0.156 + 0.987i)9-s + (0.852 + 0.522i)11-s + (0.522 + 0.852i)13-s + (0.951 − 0.309i)17-s + (0.0784 − 0.996i)19-s + (0.996 − 0.0784i)21-s + (0.987 + 0.156i)23-s + (−0.522 + 0.852i)27-s + (0.760 + 0.649i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)33-s + (−0.972 − 0.233i)37-s + (−0.156 + 0.987i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.829 + 0.558i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ 0.829 + 0.558i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.569456436 + 0.7849487428i\)
\(L(\frac12)\) \(\approx\) \(2.569456436 + 0.7849487428i\)
\(L(1)\) \(\approx\) \(1.625478167 + 0.3199071105i\)
\(L(1)\) \(\approx\) \(1.625478167 + 0.3199071105i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.760 + 0.649i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (0.852 + 0.522i)T \)
13 \( 1 + (0.522 + 0.852i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (0.0784 - 0.996i)T \)
23 \( 1 + (0.987 + 0.156i)T \)
29 \( 1 + (0.760 + 0.649i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.972 - 0.233i)T \)
41 \( 1 + (-0.156 - 0.987i)T \)
43 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (0.996 - 0.0784i)T \)
59 \( 1 + (-0.972 - 0.233i)T \)
61 \( 1 + (0.233 + 0.972i)T \)
67 \( 1 + (-0.996 - 0.0784i)T \)
71 \( 1 + (0.453 + 0.891i)T \)
73 \( 1 + (-0.987 - 0.156i)T \)
79 \( 1 + (-0.951 - 0.309i)T \)
83 \( 1 + (-0.0784 + 0.996i)T \)
89 \( 1 + (-0.156 + 0.987i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−20.38806851883541221942429492143, −19.54909685668772782036465409036, −18.92971511822521739500764154346, −18.2943129635102622166878106260, −17.62265999740714733210780448037, −16.762249559101356223858007626708, −15.780863752735630081898056461465, −14.87313017060932341749148794585, −14.49778267045373239415510988838, −13.73435444934981214559327153192, −12.82942028146345867939796480015, −12.15472095082001535910179788848, −11.521944166972963565495642405669, −10.47117732096887933981167359831, −9.54649486911196221933454636305, −8.54886696451736497804387368022, −8.306142709783841520608441685845, −7.42588924878205821993614749286, −6.33955200818941319931701687447, −5.76211578842702710236896637365, −4.65911528106517321057181578304, −3.394748536388139301615808174563, −3.01548966733334746816836402343, −1.63339449069433766408627693756, −1.16791972304019470837291741877, 1.1824682117249190383396008997, 2.01320917535910057460166015394, 3.1967756098240891699621693597, 3.97686638672616479161741896312, 4.64766369011169936303833431378, 5.43792324972490525023595391229, 6.96912821540842425185475611067, 7.26179029708428418873706046719, 8.46716836517032829611996869587, 8.99338875265878631154975537620, 9.79090731299407086366320389137, 10.60610250154700402344993463201, 11.33213463000767826196123483748, 12.10445487107772274386222611200, 13.36209540554822080428398042928, 13.871555941806651151315294532961, 14.57094914883164786456354980108, 15.12826872669713799981353242686, 16.07368663133983228517644809010, 16.80904275850149674028454047444, 17.37588667140855397919028882636, 18.415123683335854005739867223462, 19.25634253559498069686718114426, 19.86972369689889858379418575079, 20.64323388588580024585134226945

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