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Properties

Label 1-4024-4024.1219-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.208 - 0.978i$
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  + (0.999 + 0.0250i)3-s + (0.600 − 0.799i)5-s + (0.441 − 0.897i)7-s + (0.998 + 0.0500i)9-s + (0.360 − 0.932i)11-s + (0.180 + 0.983i)13-s + (0.620 − 0.784i)15-s + (−0.0437 − 0.999i)17-s + (−0.920 − 0.389i)19-s + (0.463 − 0.886i)21-s + (−0.984 + 0.174i)23-s + (−0.277 − 0.960i)25-s + (0.997 + 0.0750i)27-s + (0.496 − 0.868i)29-s + (−0.905 + 0.424i)31-s + ⋯
L(s)  = 1  + (0.999 + 0.0250i)3-s + (0.600 − 0.799i)5-s + (0.441 − 0.897i)7-s + (0.998 + 0.0500i)9-s + (0.360 − 0.932i)11-s + (0.180 + 0.983i)13-s + (0.620 − 0.784i)15-s + (−0.0437 − 0.999i)17-s + (−0.920 − 0.389i)19-s + (0.463 − 0.886i)21-s + (−0.984 + 0.174i)23-s + (−0.277 − 0.960i)25-s + (0.997 + 0.0750i)27-s + (0.496 − 0.868i)29-s + (−0.905 + 0.424i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.208 - 0.978i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.208 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.923218755 - 2.376530002i\)
\(L(\frac12)\) \(\approx\) \(1.923218755 - 2.376530002i\)
\(L(1)\) \(\approx\) \(1.616556467 - 0.6349934972i\)
\(L(1)\) \(\approx\) \(1.616556467 - 0.6349934972i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.999 + 0.0250i)T \)
5 \( 1 + (0.600 - 0.799i)T \)
7 \( 1 + (0.441 - 0.897i)T \)
11 \( 1 + (0.360 - 0.932i)T \)
13 \( 1 + (0.180 + 0.983i)T \)
17 \( 1 + (-0.0437 - 0.999i)T \)
19 \( 1 + (-0.920 - 0.389i)T \)
23 \( 1 + (-0.984 + 0.174i)T \)
29 \( 1 + (0.496 - 0.868i)T \)
31 \( 1 + (-0.905 + 0.424i)T \)
37 \( 1 + (-0.372 + 0.928i)T \)
41 \( 1 + (0.630 - 0.776i)T \)
43 \( 1 + (-0.131 + 0.991i)T \)
47 \( 1 + (0.0812 + 0.996i)T \)
53 \( 1 + (0.920 - 0.389i)T \)
59 \( 1 + (0.0687 - 0.997i)T \)
61 \( 1 + (-0.384 - 0.923i)T \)
67 \( 1 + (0.620 + 0.784i)T \)
71 \( 1 + (0.824 - 0.565i)T \)
73 \( 1 + (-0.570 + 0.821i)T \)
79 \( 1 + (0.787 + 0.615i)T \)
83 \( 1 + (-0.704 - 0.709i)T \)
89 \( 1 + (-0.289 - 0.957i)T \)
97 \( 1 + (-0.722 + 0.691i)T \)
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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

−18.44125021484163185489454472144, −18.17817299371956258401123777864, −17.58510535657444428912700775007, −16.65580018141408581692952975988, −15.58148064292211026378669179365, −14.99274058191203383045066696236, −14.7664219743343713567640756364, −14.094330092087870604847949709823, −13.21353602949567347132571151265, −12.56850259850377975533142814654, −12.05691546164853888575658827686, −10.73398073850510643184120750622, −10.43535550725935096684858211577, −9.63241729601284587874337035032, −8.89533945867923770333178118106, −8.28877558035038347429150881627, −7.56968259892482568290757926664, −6.79851291902867958878040859489, −5.99775934881585967750650549723, −5.3374353169001118501589076950, −4.180418993959944421635411714854, −3.60520447039610591849003864576, −2.577940138588665249609726953999, −2.09526552745811433389834446894, −1.49310786932658043033322976654, 0.67358963920416481123649620970, 1.55342663812868622089300849577, 2.17586338112950893953525584601, 3.179197293016364363484811281232, 4.198233101332140363077432010104, 4.418997483714741187288640307494, 5.462173705911550309545194681623, 6.48843339068789410625119212865, 7.03726046096250979673888226567, 8.15190395254839127523534062656, 8.39949117829056505175898170106, 9.3554938617809852022807647014, 9.65720592107878908884296741769, 10.64221118363410154012864737399, 11.36004369952081245029688913426, 12.20421956629062041621890305733, 13.09469257027974042611687340629, 13.680973278271362815035456920430, 14.07296611581248870539873979657, 14.53243547867357083933018851278, 15.75467278149944419634771836771, 16.197003163169844827964073923993, 16.87318786696775608397558724839, 17.54552715360621786746189076800, 18.315657697605340426626713553204

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