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Properties

Label 1-4024-4024.1043-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.961 - 0.276i$
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  + (−0.253 + 0.967i)3-s + (0.0687 + 0.997i)5-s + (0.934 + 0.355i)7-s + (−0.871 − 0.490i)9-s + (0.968 + 0.247i)11-s + (0.630 + 0.776i)13-s + (−0.982 − 0.186i)15-s + (0.0562 + 0.998i)17-s + (0.570 + 0.821i)19-s + (−0.580 + 0.814i)21-s + (−0.974 + 0.223i)23-s + (−0.990 + 0.137i)25-s + (0.695 − 0.718i)27-s + (0.217 − 0.976i)29-s + (−0.845 + 0.533i)31-s + ⋯
L(s)  = 1  + (−0.253 + 0.967i)3-s + (0.0687 + 0.997i)5-s + (0.934 + 0.355i)7-s + (−0.871 − 0.490i)9-s + (0.968 + 0.247i)11-s + (0.630 + 0.776i)13-s + (−0.982 − 0.186i)15-s + (0.0562 + 0.998i)17-s + (0.570 + 0.821i)19-s + (−0.580 + 0.814i)21-s + (−0.974 + 0.223i)23-s + (−0.990 + 0.137i)25-s + (0.695 − 0.718i)27-s + (0.217 − 0.976i)29-s + (−0.845 + 0.533i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.961 - 0.276i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.961 - 0.276i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2092702938 + 1.485045002i\)
\(L(\frac12)\) \(\approx\) \(-0.2092702938 + 1.485045002i\)
\(L(1)\) \(\approx\) \(0.8132421589 + 0.7121967643i\)
\(L(1)\) \(\approx\) \(0.8132421589 + 0.7121967643i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.253 + 0.967i)T \)
5 \( 1 + (0.0687 + 0.997i)T \)
7 \( 1 + (0.934 + 0.355i)T \)
11 \( 1 + (0.968 + 0.247i)T \)
13 \( 1 + (0.630 + 0.776i)T \)
17 \( 1 + (0.0562 + 0.998i)T \)
19 \( 1 + (0.570 + 0.821i)T \)
23 \( 1 + (-0.974 + 0.223i)T \)
29 \( 1 + (0.217 - 0.976i)T \)
31 \( 1 + (-0.845 + 0.533i)T \)
37 \( 1 + (-0.395 - 0.918i)T \)
41 \( 1 + (-0.452 - 0.891i)T \)
43 \( 1 + (0.168 - 0.985i)T \)
47 \( 1 + (-0.337 + 0.941i)T \)
53 \( 1 + (-0.570 + 0.821i)T \)
59 \( 1 + (-0.951 - 0.307i)T \)
61 \( 1 + (0.485 + 0.874i)T \)
67 \( 1 + (-0.982 + 0.186i)T \)
71 \( 1 + (0.992 + 0.124i)T \)
73 \( 1 + (-0.325 + 0.945i)T \)
79 \( 1 + (-0.265 - 0.964i)T \)
83 \( 1 + (-0.528 - 0.848i)T \)
89 \( 1 + (-0.496 + 0.868i)T \)
97 \( 1 + (-0.686 - 0.726i)T \)
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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

−18.064476007041810010511081688401, −17.552086226670695248134397787204, −16.75383714443264198806773008499, −16.36802786234372958904646941793, −15.43846517380784893189122385007, −14.42697350496514197193447054025, −13.89762470194713950928363918318, −13.32470489174186861164674876932, −12.676969535503507292692378420381, −11.75900336082563721316408722542, −11.55483455883592688993865451163, −10.74033328971397074059134234739, −9.650981753550793674747506444559, −8.89259536892553413574247121768, −8.22297991968805315396243090766, −7.74981847580837292070271041372, −6.86348983925374276499641284902, −6.14961350934749987658267504522, −5.218918766067173162906383061811, −4.88226866910721622857000603841, −3.79903051107765856407329981704, −2.85980685457224158752163754891, −1.67500353443250743540152703013, −1.26145203256010452282540047473, −0.434060157413805984050351738090, 1.50010622008164523406054229425, 2.073445759221781179217300151364, 3.27460723344569602147055043635, 3.97278853050572221737282465503, 4.34574010361248147175853063549, 5.70399500092588117331548997810, 5.87905228695493825275960523462, 6.823341087678876728369993849542, 7.71752892117201647641519573077, 8.54884853913379737912728094043, 9.19963475107322349566862054756, 9.956823449208183687008186227471, 10.64603030124649923470547464340, 11.18802249907613528051025160442, 11.853738910203196367779299148, 12.30915367671983505194551904078, 13.886977388666229848624232676773, 14.18768021724915098839364690622, 14.71378024212615394073718080111, 15.436827974812985866974867301395, 15.988859343952774126904119720764, 16.891030837064226352736133404286, 17.489971515138589046525674950026, 18.00640980034624662630019569853, 18.81601334466579427174040575013

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