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Properties

Label 1-6025-6025.289-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.0592 + 0.998i$
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  + (0.951 + 0.309i)2-s + (−0.951 + 0.309i)3-s + (0.809 + 0.587i)4-s − 6-s + (−0.891 + 0.453i)7-s + (0.587 + 0.809i)8-s + (0.809 − 0.587i)9-s + (0.891 − 0.453i)11-s + (−0.951 − 0.309i)12-s + (0.453 + 0.891i)13-s + (−0.987 + 0.156i)14-s + (0.309 + 0.951i)16-s + (−0.891 − 0.453i)17-s + (0.951 − 0.309i)18-s + (0.987 + 0.156i)19-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (−0.951 + 0.309i)3-s + (0.809 + 0.587i)4-s − 6-s + (−0.891 + 0.453i)7-s + (0.587 + 0.809i)8-s + (0.809 − 0.587i)9-s + (0.891 − 0.453i)11-s + (−0.951 − 0.309i)12-s + (0.453 + 0.891i)13-s + (−0.987 + 0.156i)14-s + (0.309 + 0.951i)16-s + (−0.891 − 0.453i)17-s + (0.951 − 0.309i)18-s + (0.987 + 0.156i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0592 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0592 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.0592 + 0.998i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.0592 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.031956670 + 1.914934984i\)
\(L(\frac12)\) \(\approx\) \(2.031956670 + 1.914934984i\)
\(L(1)\) \(\approx\) \(1.403954025 + 0.6446195688i\)
\(L(1)\) \(\approx\) \(1.403954025 + 0.6446195688i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
7 \( 1 + (-0.891 + 0.453i)T \)
11 \( 1 + (0.891 - 0.453i)T \)
13 \( 1 + (0.453 + 0.891i)T \)
17 \( 1 + (-0.891 - 0.453i)T \)
19 \( 1 + (0.987 + 0.156i)T \)
23 \( 1 + (0.987 + 0.156i)T \)
29 \( 1 + (0.951 + 0.309i)T \)
31 \( 1 + (0.891 + 0.453i)T \)
37 \( 1 + (0.156 + 0.987i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.891 + 0.453i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (-0.587 + 0.809i)T \)
61 \( 1 + (-0.587 - 0.809i)T \)
67 \( 1 + (-0.951 + 0.309i)T \)
71 \( 1 + (-0.453 - 0.891i)T \)
73 \( 1 + (0.453 - 0.891i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (-0.453 - 0.891i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−17.5187773726028577408137713164, −16.80942172830757398231023763187, −16.17222424537157627244429551502, −15.55734064139510178683532065248, −15.06376782852329486045708797154, −13.98032692119379869430200177357, −13.50279709216906809898728551193, −12.77454748413132366184428089032, −12.52996216001933074006291878383, −11.59553556010837339948174421303, −11.176980828744124647589105462345, −10.3947550858269657006880853863, −9.92409363326755348351246847774, −9.1056413345511293982385236152, −7.8535700273580055928343842270, −7.10392973929093277945950444225, −6.54092834977793187801859426700, −6.1052914017302758206526622114, −5.32926977861689890545205557339, −4.51351527050431446781281963594, −3.99777282824942850968549343982, −3.118806247528067075803097509334, −2.35571634061651123295264536726, −1.202362820598085729853972418654, −0.77163667612926467888151547557, 0.90486222962250177934447182346, 1.83052693584374471465960446862, 3.08931037129944870952205155930, 3.39528153118113298948276987674, 4.520811026064684257852487091780, 4.74734240428290968560671423221, 5.86979455415929672344233169035, 6.20538068897230496505603859911, 6.82983035323636798871297065606, 7.32935576154800452927987399208, 8.73637454709846708231957948568, 9.09753061375201211894787095639, 10.03396114395195429022044231567, 10.83266727012307572312057555125, 11.573648597267314480319019656, 11.89105906675767183367974431272, 12.48313187037429638628393939819, 13.43466107541559052845235095901, 13.73362321404110313087804225646, 14.643550732796539469355240834513, 15.49728732009476158835236771132, 15.83772620602725834299789028162, 16.475076597006579424641787284492, 16.878410971902934026211675494583, 17.6973333549839272821215600802

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