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Properties

Label 1-525-525.317-r0-0-0
Degree $1$
Conductor $525$
Sign $0.796 + 0.605i$
Analytic cond. $2.43808$
Root an. cond. $2.43808$
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  + (0.406 − 0.913i)2-s + (−0.669 − 0.743i)4-s + (−0.951 + 0.309i)8-s + (0.104 + 0.994i)11-s + (−0.587 + 0.809i)13-s + (−0.104 + 0.994i)16-s + (−0.207 + 0.978i)17-s + (−0.669 + 0.743i)19-s + (0.951 + 0.309i)22-s + (−0.406 + 0.913i)23-s + (0.5 + 0.866i)26-s + (0.309 − 0.951i)29-s + (−0.978 − 0.207i)31-s + (0.866 + 0.5i)32-s + (0.809 + 0.587i)34-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)2-s + (−0.669 − 0.743i)4-s + (−0.951 + 0.309i)8-s + (0.104 + 0.994i)11-s + (−0.587 + 0.809i)13-s + (−0.104 + 0.994i)16-s + (−0.207 + 0.978i)17-s + (−0.669 + 0.743i)19-s + (0.951 + 0.309i)22-s + (−0.406 + 0.913i)23-s + (0.5 + 0.866i)26-s + (0.309 − 0.951i)29-s + (−0.978 − 0.207i)31-s + (0.866 + 0.5i)32-s + (0.809 + 0.587i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.796 + 0.605i$
Analytic conductor: \(2.43808\)
Root analytic conductor: \(2.43808\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 525,\ (0:\ ),\ 0.796 + 0.605i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9044591109 + 0.3046576922i\)
\(L(\frac12)\) \(\approx\) \(0.9044591109 + 0.3046576922i\)
\(L(1)\) \(\approx\) \(0.9614026718 - 0.2296699314i\)
\(L(1)\) \(\approx\) \(0.9614026718 - 0.2296699314i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.406 - 0.913i)T \)
11 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (-0.207 + 0.978i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (-0.406 + 0.913i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.994 - 0.104i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.207 + 0.978i)T \)
53 \( 1 + (-0.743 + 0.669i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (0.207 - 0.978i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.994 + 0.104i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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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

−23.58555368201105138010019059001, −22.50855650887284795673326470961, −22.05921866212956282922853904346, −21.13663295378101884010265960399, −20.13067955182513416538290283472, −19.07874794877475313453766175729, −18.09680085247632151593315996674, −17.43799149767290448921935437934, −16.39159051578307075596361590211, −15.90008386156080378331666078383, −14.79840131508226191362222127510, −14.18073156630120750853299707737, −13.21137145511129688506673687445, −12.49802106354527468119375302279, −11.425038666997841935948301106585, −10.327931647847085596844512785669, −9.06679500877762227175428843285, −8.435582591348973015645292688481, −7.356689544424841941669086453463, −6.561030983362367014945178327696, −5.502423008884221182824269708097, −4.75145340043414766473730020483, −3.53556371957222218828424579613, −2.58426437520175921566901992298, −0.43065536312150020334019781064, 1.62840312503377156685139992666, 2.289706288850204350557433693926, 3.78722265501346854851624587190, 4.39028165129737407823201132133, 5.54138552198667708712792571438, 6.54765935730347424585511265421, 7.792734146435970372560032870642, 9.0097850681973005920444247943, 9.80516883133079179105764497075, 10.57938857612228567246356784538, 11.62914660620924678824170746266, 12.36588635976992021499501256248, 13.071608772897996838728706094535, 14.18812276499677634791567960760, 14.79104327857880249259557679662, 15.71305942087585877273855459223, 17.12692932938441092644979729788, 17.68934950816407346244254354854, 18.89448467355959503807625336094, 19.40910924854210284174258352565, 20.29226920555795537928324074183, 21.11959496712393844288191890087, 21.834113000349263282071599144822, 22.621757060621126678825389523729, 23.51085865795540654178928864022

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