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Properties

Label 1-4024-4024.1347-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.517 - 0.855i$
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  + (−0.395 − 0.918i)3-s + (−0.610 − 0.791i)5-s + (0.999 + 0.0375i)7-s + (−0.686 + 0.726i)9-s + (0.780 + 0.625i)11-s + (−0.980 − 0.198i)13-s + (−0.485 + 0.874i)15-s + (−0.313 + 0.949i)17-s + (−0.974 + 0.223i)19-s + (−0.360 − 0.932i)21-s + (−0.289 − 0.957i)23-s + (−0.253 + 0.967i)25-s + (0.939 + 0.343i)27-s + (0.192 + 0.981i)29-s + (0.998 + 0.0500i)31-s + ⋯
L(s)  = 1  + (−0.395 − 0.918i)3-s + (−0.610 − 0.791i)5-s + (0.999 + 0.0375i)7-s + (−0.686 + 0.726i)9-s + (0.780 + 0.625i)11-s + (−0.980 − 0.198i)13-s + (−0.485 + 0.874i)15-s + (−0.313 + 0.949i)17-s + (−0.974 + 0.223i)19-s + (−0.360 − 0.932i)21-s + (−0.289 − 0.957i)23-s + (−0.253 + 0.967i)25-s + (0.939 + 0.343i)27-s + (0.192 + 0.981i)29-s + (0.998 + 0.0500i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.517 - 0.855i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.517 - 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4705684927 - 0.8347712173i\)
\(L(\frac12)\) \(\approx\) \(0.4705684927 - 0.8347712173i\)
\(L(1)\) \(\approx\) \(0.7669013740 - 0.3115839058i\)
\(L(1)\) \(\approx\) \(0.7669013740 - 0.3115839058i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.395 - 0.918i)T \)
5 \( 1 + (-0.610 - 0.791i)T \)
7 \( 1 + (0.999 + 0.0375i)T \)
11 \( 1 + (0.780 + 0.625i)T \)
13 \( 1 + (-0.980 - 0.198i)T \)
17 \( 1 + (-0.313 + 0.949i)T \)
19 \( 1 + (-0.974 + 0.223i)T \)
23 \( 1 + (-0.289 - 0.957i)T \)
29 \( 1 + (0.192 + 0.981i)T \)
31 \( 1 + (0.998 + 0.0500i)T \)
37 \( 1 + (-0.951 + 0.307i)T \)
41 \( 1 + (0.463 + 0.886i)T \)
43 \( 1 + (-0.817 - 0.575i)T \)
47 \( 1 + (-0.143 + 0.989i)T \)
53 \( 1 + (0.974 + 0.223i)T \)
59 \( 1 + (0.747 - 0.663i)T \)
61 \( 1 + (-0.265 - 0.964i)T \)
67 \( 1 + (-0.485 - 0.874i)T \)
71 \( 1 + (-0.943 - 0.331i)T \)
73 \( 1 + (-0.739 - 0.673i)T \)
79 \( 1 + (-0.539 - 0.842i)T \)
83 \( 1 + (-0.858 - 0.512i)T \)
89 \( 1 + (-0.947 + 0.319i)T \)
97 \( 1 + (0.810 + 0.585i)T \)
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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

−18.700039580095113822523472221451, −17.79576597068296850673422175178, −17.354712463465668084950410091609, −16.73020356381252891372895283151, −15.83667327869872580289997160358, −15.323648263512334643345653234942, −14.664500774996105210664612233522, −14.22133177704315298635839082768, −13.47343126050614061455822146636, −12.0148210468599949404040905334, −11.649102615824614044866587851883, −11.35648346260982026152896674551, −10.37503120090191137060157098316, −9.97113483714845177136102167917, −8.894812372027555026501533826559, −8.4438128776206823762108083987, −7.39715600896845529445606998362, −6.839636511054819500595660582947, −5.92629849548264050550192011043, −5.1616280642037401707664689482, −4.2869721767639501466824943494, −3.976736306260000950209801293410, −2.90783848381443039700634178192, −2.20258307141476935986417467994, −0.799011789384758004168071141492, 0.366978528132468073976054415991, 1.59236185722126754656745928729, 1.80154090166627770646848845723, 2.99745397356668581156557709987, 4.422531916637050647361600883395, 4.53443513792259618806987813415, 5.46478671416858390649428235385, 6.38349397071594842561833710451, 7.04967724793914590819876186869, 7.81731374741699089057238942890, 8.42991190814090319853887554892, 8.839766471534950139876168250866, 10.09510016598547716044928361574, 10.80649536406786107411364112639, 11.59475452274312544878082059346, 12.26971623727407763008392796665, 12.46239128222514212190483314104, 13.27255730641800889486233500902, 14.27956526142428576201239322222, 14.746180415448119670804002703389, 15.40723435247663375035239482412, 16.53024163478527297549885869813, 17.03013419420581151724529209550, 17.477959711362198529410614401379, 18.08367497535205972952302242618

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