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Properties

Label 1-4024-4024.1267-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.181 + 0.983i$
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  + (0.713 − 0.700i)3-s + (−0.905 − 0.424i)5-s + (−0.999 + 0.0250i)7-s + (0.0187 − 0.999i)9-s + (0.900 − 0.435i)11-s + (0.610 + 0.791i)13-s + (−0.943 + 0.331i)15-s + (0.977 − 0.211i)17-s + (−0.988 − 0.149i)19-s + (−0.695 + 0.718i)21-s + (−0.659 + 0.752i)23-s + (0.640 + 0.768i)25-s + (−0.686 − 0.726i)27-s + (0.384 + 0.923i)29-s + (−0.528 − 0.848i)31-s + ⋯
L(s)  = 1  + (0.713 − 0.700i)3-s + (−0.905 − 0.424i)5-s + (−0.999 + 0.0250i)7-s + (0.0187 − 0.999i)9-s + (0.900 − 0.435i)11-s + (0.610 + 0.791i)13-s + (−0.943 + 0.331i)15-s + (0.977 − 0.211i)17-s + (−0.988 − 0.149i)19-s + (−0.695 + 0.718i)21-s + (−0.659 + 0.752i)23-s + (0.640 + 0.768i)25-s + (−0.686 − 0.726i)27-s + (0.384 + 0.923i)29-s + (−0.528 − 0.848i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.181 + 0.983i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.181 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1631942710 + 0.1960762855i\)
\(L(\frac12)\) \(\approx\) \(0.1631942710 + 0.1960762855i\)
\(L(1)\) \(\approx\) \(0.8641762014 - 0.2592752627i\)
\(L(1)\) \(\approx\) \(0.8641762014 - 0.2592752627i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.713 - 0.700i)T \)
5 \( 1 + (-0.905 - 0.424i)T \)
7 \( 1 + (-0.999 + 0.0250i)T \)
11 \( 1 + (0.900 - 0.435i)T \)
13 \( 1 + (0.610 + 0.791i)T \)
17 \( 1 + (0.977 - 0.211i)T \)
19 \( 1 + (-0.988 - 0.149i)T \)
23 \( 1 + (-0.659 + 0.752i)T \)
29 \( 1 + (0.384 + 0.923i)T \)
31 \( 1 + (-0.528 - 0.848i)T \)
37 \( 1 + (-0.668 + 0.743i)T \)
41 \( 1 + (-0.747 + 0.663i)T \)
43 \( 1 + (-0.803 - 0.595i)T \)
47 \( 1 + (-0.580 - 0.814i)T \)
53 \( 1 + (0.988 - 0.149i)T \)
59 \( 1 + (-0.845 + 0.533i)T \)
61 \( 1 + (-0.337 - 0.941i)T \)
67 \( 1 + (-0.943 - 0.331i)T \)
71 \( 1 + (0.974 - 0.223i)T \)
73 \( 1 + (-0.0312 + 0.999i)T \)
79 \( 1 + (-0.143 - 0.989i)T \)
83 \( 1 + (-0.772 - 0.635i)T \)
89 \( 1 + (0.301 + 0.953i)T \)
97 \( 1 + (-0.106 + 0.994i)T \)
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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

−18.55589237445720374692416463508, −17.56053649021779467660607668047, −16.51379325247053521299264920009, −16.32655269130262545901395478383, −15.35538991358627405853200560435, −15.09415626297915856152402927007, −14.30412825903691116019768303457, −13.70510937504760763202248332666, −12.621745548502449983332615887203, −12.31580930021670064090538557950, −11.297659958996974239466067090230, −10.35729096643640970419037271415, −10.22807452961395392813559424073, −9.23599641450411545162194205262, −8.49951662619053478460432031969, −7.99069270861111308245154330392, −7.09679658551080775007771403579, −6.40724258669968621239156290071, −5.56293363358494069265264618815, −4.384642230665431283207795626917, −3.89494894654653731627877631571, −3.30501752305462585438437687784, −2.65620626265694919695106681419, −1.51566186035096806375035001133, −0.06768027595177424360954414018, 1.1102467902723555886958936709, 1.78123811964524323016740802107, 3.00601846795683367526203900195, 3.66950755899453403234085937504, 3.97628257863316371294297039838, 5.24622664379658668877364172165, 6.346984899007719067706725385656, 6.68286258803826621425854805626, 7.50517127490679731493680942016, 8.30501139926985740613843709194, 8.857673350869942548886526364222, 9.39261600793060575694810268208, 10.26781642849056266164687219750, 11.40280721280407067282920830414, 11.97514556980061780733514914381, 12.405627433027123760039502766, 13.349363884231584290260397105, 13.676627656515058576060003576404, 14.62708356880135473441595909434, 15.17332946133283444040738665326, 15.987143540616163181393228749476, 16.63422972947157428686344661333, 17.07792685926850447867079634731, 18.38223651240859693157159897795, 18.78142969285569154023323976035

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