Properties

Label 1-6025-6025.5522-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.312 + 0.949i$
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  + (−0.156 − 0.987i)2-s + (−0.156 − 0.987i)3-s + (−0.951 + 0.309i)4-s + (−0.951 + 0.309i)6-s + (−0.996 − 0.0784i)7-s + (0.453 + 0.891i)8-s + (−0.951 + 0.309i)9-s + (0.522 + 0.852i)11-s + (0.453 + 0.891i)12-s + (−0.996 + 0.0784i)13-s + (0.0784 + 0.996i)14-s + (0.809 − 0.587i)16-s + (0.382 + 0.923i)17-s + (0.453 + 0.891i)18-s + (−0.760 − 0.649i)19-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)2-s + (−0.156 − 0.987i)3-s + (−0.951 + 0.309i)4-s + (−0.951 + 0.309i)6-s + (−0.996 − 0.0784i)7-s + (0.453 + 0.891i)8-s + (−0.951 + 0.309i)9-s + (0.522 + 0.852i)11-s + (0.453 + 0.891i)12-s + (−0.996 + 0.0784i)13-s + (0.0784 + 0.996i)14-s + (0.809 − 0.587i)16-s + (0.382 + 0.923i)17-s + (0.453 + 0.891i)18-s + (−0.760 − 0.649i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.312 + 0.949i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (5522, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.312 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02053548614 + 0.02837928270i\)
\(L(\frac12)\) \(\approx\) \(0.02053548614 + 0.02837928270i\)
\(L(1)\) \(\approx\) \(0.5166716071 - 0.3340357719i\)
\(L(1)\) \(\approx\) \(0.5166716071 - 0.3340357719i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.156 - 0.987i)T \)
3 \( 1 + (-0.156 - 0.987i)T \)
7 \( 1 + (-0.996 - 0.0784i)T \)
11 \( 1 + (0.522 + 0.852i)T \)
13 \( 1 + (-0.996 + 0.0784i)T \)
17 \( 1 + (0.382 + 0.923i)T \)
19 \( 1 + (-0.760 - 0.649i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + (-0.923 + 0.382i)T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (0.987 - 0.156i)T \)
43 \( 1 + (0.0784 + 0.996i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + (0.923 - 0.382i)T \)
73 \( 1 + (-0.996 - 0.0784i)T \)
79 \( 1 + (0.453 + 0.891i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.972 - 0.233i)T \)
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

−17.060286263828711463404028416327, −16.75539114129316973305042851045, −16.247820775613013446903960258799, −15.77002866772242703509892966541, −14.80310802514768518758683586946, −14.61083955840303081168311438188, −13.797271314257086768685239485450, −13.105980088881327393251346631468, −12.20169305793628308535165791307, −11.648952997092736415896478477930, −10.488686672413141156370846918398, −10.07153260944647756638246644873, −9.51590761700416461953928325807, −8.82986505755549416444209641756, −8.30793124027038856156710376405, −7.34489989033064781495561470364, −6.58769967736827897363746810298, −5.95238742648469713333911621653, −5.47728733598309594843548191660, −4.53994808421521284474475663448, −4.007253414107978978186208448204, −3.22816003077300542146347544274, −2.42491884293430987345050638730, −0.7574185464810386086821548638, −0.01411096724898980971575109529, 1.10832931116395780673830128777, 1.864816596132755565091185680548, 2.50670055293513776858097277232, 3.25407697395095827040874457566, 4.06502280895722091092378172086, 4.86453677503233803149557403185, 5.75495557220732548421331923933, 6.49298279104449576961016874690, 7.28360695653563823414233728167, 7.766339815048369663790526662173, 8.75546031210374822805833617990, 9.351435896111252783530712788221, 9.92586500598761504177987165301, 10.73712331709670976153533965157, 11.35052679493729481632491234978, 12.23956455214277999187093628939, 12.69360501357632858556736980525, 12.803563752502005814510957816173, 13.82794235029377583070776347454, 14.41675511918936529796668745166, 15.03431084675430377969271343585, 16.20471496160095048155640731141, 16.92841524365543685230906445005, 17.4401667075475137844832498252, 17.92065650423054764475793750322

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