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Properties

Label 1-4013-4013.56-r1-0-0
Degree $1$
Conductor $4013$
Sign $-0.338 + 0.940i$
Analytic cond. $431.256$
Root an. cond. $431.256$
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  + (−0.578 − 0.815i)2-s + (−0.735 + 0.677i)3-s + (−0.330 + 0.943i)4-s + (−0.472 − 0.881i)5-s + (0.978 + 0.208i)6-s + (−0.996 + 0.0782i)7-s + (0.960 − 0.276i)8-s + (0.0829 − 0.996i)9-s + (−0.445 + 0.895i)10-s + (−0.321 − 0.946i)11-s + (−0.396 − 0.918i)12-s + (0.151 − 0.988i)13-s + (0.640 + 0.767i)14-s + (0.944 + 0.328i)15-s + (−0.781 − 0.623i)16-s + (−0.982 + 0.186i)17-s + ⋯
L(s)  = 1  + (−0.578 − 0.815i)2-s + (−0.735 + 0.677i)3-s + (−0.330 + 0.943i)4-s + (−0.472 − 0.881i)5-s + (0.978 + 0.208i)6-s + (−0.996 + 0.0782i)7-s + (0.960 − 0.276i)8-s + (0.0829 − 0.996i)9-s + (−0.445 + 0.895i)10-s + (−0.321 − 0.946i)11-s + (−0.396 − 0.918i)12-s + (0.151 − 0.988i)13-s + (0.640 + 0.767i)14-s + (0.944 + 0.328i)15-s + (−0.781 − 0.623i)16-s + (−0.982 + 0.186i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4013\)
Sign: $-0.338 + 0.940i$
Analytic conductor: \(431.256\)
Root analytic conductor: \(431.256\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4013} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4013,\ (1:\ ),\ -0.338 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1704358442 - 0.2424771804i\)
\(L(\frac12)\) \(\approx\) \(-0.1704358442 - 0.2424771804i\)
\(L(1)\) \(\approx\) \(0.3884981713 - 0.2061241186i\)
\(L(1)\) \(\approx\) \(0.3884981713 - 0.2061241186i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4013 \( 1 \)
good2 \( 1 + (-0.578 - 0.815i)T \)
3 \( 1 + (-0.735 + 0.677i)T \)
5 \( 1 + (-0.472 - 0.881i)T \)
7 \( 1 + (-0.996 + 0.0782i)T \)
11 \( 1 + (-0.321 - 0.946i)T \)
13 \( 1 + (0.151 - 0.988i)T \)
17 \( 1 + (-0.982 + 0.186i)T \)
19 \( 1 + (0.527 + 0.849i)T \)
23 \( 1 + (0.608 - 0.793i)T \)
29 \( 1 + (-0.369 + 0.929i)T \)
31 \( 1 + (-0.695 + 0.718i)T \)
37 \( 1 + (-0.990 - 0.135i)T \)
41 \( 1 + (0.960 + 0.278i)T \)
43 \( 1 + (0.751 - 0.659i)T \)
47 \( 1 + (0.145 - 0.989i)T \)
53 \( 1 + (-0.850 - 0.526i)T \)
59 \( 1 + (-0.148 - 0.988i)T \)
61 \( 1 + (0.963 + 0.266i)T \)
67 \( 1 + (0.681 + 0.731i)T \)
71 \( 1 + (-0.880 - 0.474i)T \)
73 \( 1 + (-0.200 + 0.979i)T \)
79 \( 1 + (0.0344 - 0.999i)T \)
83 \( 1 + (0.434 - 0.900i)T \)
89 \( 1 + (0.991 + 0.128i)T \)
97 \( 1 + (-0.990 - 0.137i)T \)
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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

−18.73645699524589623151959509597, −17.95370860664321478369905846564, −17.55283376835564823242398497172, −16.84629593873941064061663322781, −15.86385606217572133558960306021, −15.75039738902603570260516585266, −14.92149148952187180388771322197, −13.94503062117028095639067147293, −13.44331787048677588782229662362, −12.73179010575687380302943201937, −11.75110164105677476222935231885, −11.08135345455341019301626327468, −10.63056893669891924875778059275, −9.5028530539533367976512434099, −9.31642740712689161766178137549, −7.9946914578052461300525298155, −7.20908407255195072978416580488, −7.07305499658601937255591882783, −6.35676945735102806498191569119, −5.70719444941982224402663042237, −4.691263734565899169393084182142, −4.03410422238271570758669511184, −2.65806548148691747735543166821, −1.988130525463658439993120539332, −0.82887633577219814986771781852, 0.15169612967436477902244572665, 0.48580314293452150721932527167, 1.43235379984299369449458144180, 2.80567847549378047751513350331, 3.54806315586991811858410090098, 3.93612554847105994634460448439, 5.067599385042881136637623628148, 5.531810131521384709220659974219, 6.562591249447660084488070962534, 7.44198882994630476958829845365, 8.48110339422392257676365694478, 8.87328256639155868846748956047, 9.52480163006594419051793499031, 10.48159486711909577639295640498, 10.73090938349099635454912961771, 11.56062965287660052192351120068, 12.32439701991969530682128242155, 12.82135158941032024937662333315, 13.247491135038678293742604389674, 14.44266129080967575939569304577, 15.59330172872929067997671828407, 16.147821012767119836851095796375, 16.31924591263905162432718744194, 17.19204690072282314987390925310, 17.76086347757158092476076820680

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