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Properties

Label 1-6025-6025.6019-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.998 + 0.0492i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (−0.809 + 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s + (0.309 − 0.951i)6-s + (−0.951 − 0.309i)7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.587 − 0.809i)11-s + (0.309 + 0.951i)12-s + (−0.951 + 0.309i)13-s + (0.951 − 0.309i)14-s + (−0.809 − 0.587i)16-s + i·17-s + (0.309 + 0.951i)18-s + (−0.951 + 0.309i)19-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s + (0.309 − 0.951i)6-s + (−0.951 − 0.309i)7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.587 − 0.809i)11-s + (0.309 + 0.951i)12-s + (−0.951 + 0.309i)13-s + (0.951 − 0.309i)14-s + (−0.809 − 0.587i)16-s + i·17-s + (0.309 + 0.951i)18-s + (−0.951 + 0.309i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.998 + 0.0492i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (6019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.998 + 0.0492i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001250147885 - 0.05076344916i\)
\(L(\frac12)\) \(\approx\) \(0.001250147885 - 0.05076344916i\)
\(L(1)\) \(\approx\) \(0.3805633793 + 0.07023565362i\)
\(L(1)\) \(\approx\) \(0.3805633793 + 0.07023565362i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + iT \)
19 \( 1 + (-0.951 + 0.309i)T \)
23 \( 1 - iT \)
29 \( 1 - T \)
31 \( 1 - iT \)
37 \( 1 - iT \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - iT \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (0.587 + 0.809i)T \)
97 \( 1 - T \)
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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

−18.16836974766682921102767336584, −17.36075364338704168688626542705, −17.035840760899624644798927855903, −16.27416908422465141527067613, −15.638824101541844133916313108144, −15.06997784133767665564274804557, −13.743642735124315480337533735189, −13.07586494147138078798121057064, −12.66627129910266921723746597709, −12.06030103322038369241607204456, −11.53548747977276037365116999527, −10.68750193904382934297027243316, −10.09200997006264318190536289601, −9.596453787509703172299367887541, −8.84244236617220460903261780878, −7.8819424017827913366608710519, −7.213057836950157504699395033256, −6.93488636083280576728472970421, −5.985184746978309168196787593164, −5.13522498274211110120850453249, −4.459041021516571924394029033750, −3.31655008262939954365393170987, −2.54378751732814862804393293586, −2.05055976085715917219514037455, −1.01123267741005347236195179693, 0.03822554005240922650826495131, 0.536791183839162676668975104850, 1.855814824347739770825209854, 2.72530510570078687296267545622, 3.84487114353818505800389978097, 4.434751156063855960443079562528, 5.433453447116159090803204133640, 6.000845408327038973283776792424, 6.43688508932198503721840722823, 7.24992672232115954807335523600, 7.94106596749455221995504812346, 8.903243669746266831192966769904, 9.34933431410726956802198826915, 10.16205147813925275667533709047, 10.62253188063308308615024798113, 11.01126189819611463216371338922, 12.05972433755721203237387462022, 12.68097734688466111875219041378, 13.43322096743336378880893201684, 14.43307883967319483849068626684, 15.0190661984032964857596559603, 15.540109078778229325633070008250, 16.39452791234371555723865121095, 16.730908098785101388225487395797, 17.04227721295062506597218530215

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