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Properties

Label 1-6041-6041.356-r0-0-0
Degree $1$
Conductor $6041$
Sign $0.877 + 0.479i$
Analytic cond. $28.0542$
Root an. cond. $28.0542$
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  + (0.982 + 0.188i)2-s + (0.206 − 0.978i)3-s + (0.929 + 0.370i)4-s + (0.0109 − 0.999i)5-s + (0.386 − 0.922i)6-s + (0.842 + 0.538i)8-s + (−0.914 − 0.403i)9-s + (0.199 − 0.979i)10-s + (−0.994 − 0.101i)11-s + (0.553 − 0.832i)12-s + (−0.883 + 0.469i)13-s + (−0.976 − 0.216i)15-s + (0.726 + 0.687i)16-s + (−0.765 + 0.644i)17-s + (−0.822 − 0.568i)18-s + (−0.141 + 0.989i)19-s + ⋯
L(s)  = 1  + (0.982 + 0.188i)2-s + (0.206 − 0.978i)3-s + (0.929 + 0.370i)4-s + (0.0109 − 0.999i)5-s + (0.386 − 0.922i)6-s + (0.842 + 0.538i)8-s + (−0.914 − 0.403i)9-s + (0.199 − 0.979i)10-s + (−0.994 − 0.101i)11-s + (0.553 − 0.832i)12-s + (−0.883 + 0.469i)13-s + (−0.976 − 0.216i)15-s + (0.726 + 0.687i)16-s + (−0.765 + 0.644i)17-s + (−0.822 − 0.568i)18-s + (−0.141 + 0.989i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6041\)    =    \(7 \cdot 863\)
Sign: $0.877 + 0.479i$
Analytic conductor: \(28.0542\)
Root analytic conductor: \(28.0542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6041} (356, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6041,\ (0:\ ),\ 0.877 + 0.479i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.370555454 + 0.6055209788i\)
\(L(\frac12)\) \(\approx\) \(2.370555454 + 0.6055209788i\)
\(L(1)\) \(\approx\) \(1.668388712 - 0.2962327947i\)
\(L(1)\) \(\approx\) \(1.668388712 - 0.2962327947i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
863 \( 1 \)
good2 \( 1 + (0.982 + 0.188i)T \)
3 \( 1 + (0.206 - 0.978i)T \)
5 \( 1 + (0.0109 - 0.999i)T \)
11 \( 1 + (-0.994 - 0.101i)T \)
13 \( 1 + (-0.883 + 0.469i)T \)
17 \( 1 + (-0.765 + 0.644i)T \)
19 \( 1 + (-0.141 + 0.989i)T \)
23 \( 1 + (-0.227 - 0.973i)T \)
29 \( 1 + (0.989 + 0.145i)T \)
31 \( 1 + (0.796 + 0.604i)T \)
37 \( 1 + (0.684 + 0.728i)T \)
41 \( 1 + (0.541 + 0.840i)T \)
43 \( 1 + (-0.983 + 0.181i)T \)
47 \( 1 + (-0.262 + 0.964i)T \)
53 \( 1 + (0.858 - 0.513i)T \)
59 \( 1 + (-0.997 - 0.0728i)T \)
61 \( 1 + (0.946 - 0.322i)T \)
67 \( 1 + (0.491 + 0.870i)T \)
71 \( 1 + (-0.465 - 0.884i)T \)
73 \( 1 + (0.879 + 0.475i)T \)
79 \( 1 + (0.290 - 0.956i)T \)
83 \( 1 + (-0.994 - 0.109i)T \)
89 \( 1 + (0.911 + 0.410i)T \)
97 \( 1 + (0.213 + 0.976i)T \)
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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

−17.594843244935593620924019785267, −16.88852195085501765300807238955, −15.80330273143578845008533282904, −15.579117493829022957321610257965, −15.119457588126358845471883604800, −14.45060870080006413583250890820, −13.65426695282723281871450695184, −13.42247386015654253299848233413, −12.35939012070996990094867019419, −11.49750671458763016013095638243, −11.17279253002713014129553236714, −10.3265676264245907033190883293, −10.051076725394670879714844149528, −9.26048007564911915014934049269, −8.12421007591226837998979433049, −7.43799517088214762141926084262, −6.82331815919362358257212185091, −5.91547734285599264863076505148, −5.244831864332295963546377568608, −4.69335741402959095230397799568, −3.9544183353543975290692098635, −3.1354389003693318604170368384, −2.493037708142446754631677074009, −2.280371525366795969972619255, −0.39738582198033841967657999036, 1.00493296787151707383586851238, 1.87988082764544301640422031943, 2.461378478871555238470867984752, 3.198556307471382239324038813105, 4.32093376668872977747979962090, 4.770134183624084077947878748923, 5.53926143380302599479013322565, 6.34574764778975123137284241925, 6.72575219007964259425739276863, 7.88909743615521145671550218590, 8.06315841089427566999054846454, 8.74587420734631887210455062716, 9.86800862860129199238442991610, 10.60154033680797409935337313095, 11.58013697718824628929916244756, 12.09422506204344034406685875632, 12.66772067454844733331513855785, 13.07427991184448750438100576650, 13.6828461693651729768968100434, 14.40511009561267592317589436686, 14.91588909844302116568505846728, 15.82009808200596839016979723853, 16.387216228101366889345279614485, 17.020963210924638792455909160796, 17.606712318051186907898628142504

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