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Properties

Label 1-4024-4024.1277-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.704 - 0.709i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
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  + (−0.0687 − 0.997i)3-s + (−0.560 − 0.828i)5-s + (−0.649 + 0.760i)7-s + (−0.990 + 0.137i)9-s + (0.986 + 0.161i)11-s + (0.106 − 0.994i)13-s + (−0.787 + 0.615i)15-s + (−0.871 − 0.490i)17-s + (−0.452 + 0.891i)19-s + (0.803 + 0.595i)21-s + (−0.463 + 0.886i)23-s + (−0.372 + 0.928i)25-s + (0.205 + 0.978i)27-s + (−0.968 + 0.247i)29-s + (0.407 − 0.913i)31-s + ⋯
L(s)  = 1  + (−0.0687 − 0.997i)3-s + (−0.560 − 0.828i)5-s + (−0.649 + 0.760i)7-s + (−0.990 + 0.137i)9-s + (0.986 + 0.161i)11-s + (0.106 − 0.994i)13-s + (−0.787 + 0.615i)15-s + (−0.871 − 0.490i)17-s + (−0.452 + 0.891i)19-s + (0.803 + 0.595i)21-s + (−0.463 + 0.886i)23-s + (−0.372 + 0.928i)25-s + (0.205 + 0.978i)27-s + (−0.968 + 0.247i)29-s + (0.407 − 0.913i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.704 - 0.709i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.704 - 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8591246987 - 0.3573776608i\)
\(L(\frac12)\) \(\approx\) \(0.8591246987 - 0.3573776608i\)
\(L(1)\) \(\approx\) \(0.7397649543 - 0.2766313231i\)
\(L(1)\) \(\approx\) \(0.7397649543 - 0.2766313231i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.0687 - 0.997i)T \)
5 \( 1 + (-0.560 - 0.828i)T \)
7 \( 1 + (-0.649 + 0.760i)T \)
11 \( 1 + (0.986 + 0.161i)T \)
13 \( 1 + (0.106 - 0.994i)T \)
17 \( 1 + (-0.871 - 0.490i)T \)
19 \( 1 + (-0.452 + 0.891i)T \)
23 \( 1 + (-0.463 + 0.886i)T \)
29 \( 1 + (-0.968 + 0.247i)T \)
31 \( 1 + (0.407 - 0.913i)T \)
37 \( 1 + (0.610 + 0.791i)T \)
41 \( 1 + (-0.996 - 0.0875i)T \)
43 \( 1 + (0.0312 - 0.999i)T \)
47 \( 1 + (0.168 + 0.985i)T \)
53 \( 1 + (-0.452 - 0.891i)T \)
59 \( 1 + (0.549 + 0.835i)T \)
61 \( 1 + (-0.0937 - 0.995i)T \)
67 \( 1 + (0.787 + 0.615i)T \)
71 \( 1 + (-0.0812 + 0.996i)T \)
73 \( 1 + (0.289 - 0.957i)T \)
79 \( 1 + (-0.507 + 0.861i)T \)
83 \( 1 + (0.982 + 0.186i)T \)
89 \( 1 + (0.360 + 0.932i)T \)
97 \( 1 + (-0.253 - 0.967i)T \)
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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

−18.70568602749274405754358464621, −17.694680342327873038361402610419, −17.070297332380564507660555392588, −16.42485367146796933176808453383, −15.925682942310360406195080210612, −15.12730487668884114513941319853, −14.57726689154574407381106171478, −13.983176820926509315588964288, −13.25384355098669457338736531909, −12.17885719339500737473805388137, −11.46352181678883287926205380265, −10.92618988233354691673166800858, −10.41842294547387555505198098361, −9.56325115168361799069996887537, −8.9862334189627455230053876816, −8.25716522191263439938586315926, −7.148730809192598070971403637024, −6.51559248284915013660412767962, −6.17385165309772325769670281516, −4.734489826174235916814749610697, −4.13769823390728998984563188257, −3.74204679749654630497123727823, −2.90860502543065161268171905314, −1.975637831132916354835944365226, −0.45650860446343046427794217217, 0.57165722674618745814728134588, 1.5526378707463247178075832251, 2.28193349286311231586577167469, 3.322611530372598263715104220851, 3.96785139958351984038972786285, 5.10254859224778581931428722352, 5.76752518515805050845038096227, 6.38967840995713064289788812716, 7.19814013397683818472179101262, 8.005199003749194284894483838758, 8.520809005464271871109585713897, 9.24110667919286828649465122747, 9.88850896433132478415764002075, 11.170981806618046055285284101056, 11.72354231874509440144009422906, 12.23459416175755546841638482755, 12.94163087245994631481953011064, 13.28254902895744924354981421459, 14.21866543239063596420943733775, 15.16715834959044975641135659045, 15.5481760807362103716563238936, 16.51988864682442449656506301266, 17.0311970423713701285103603573, 17.7123780430797790972514953589, 18.48417568151882736900577836697

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