Properties

Label 1-407-407.53-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.960 + 0.276i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (−0.241 − 0.970i)2-s + (−0.374 − 0.927i)3-s + (−0.882 + 0.469i)4-s + (0.961 − 0.275i)5-s + (−0.809 + 0.587i)6-s + (0.0348 − 0.999i)7-s + (0.669 + 0.743i)8-s + (−0.719 + 0.694i)9-s + (−0.5 − 0.866i)10-s + (0.766 + 0.642i)12-s + (0.438 − 0.898i)13-s + (−0.978 + 0.207i)14-s + (−0.615 − 0.788i)15-s + (0.559 − 0.829i)16-s + (0.438 + 0.898i)17-s + (0.848 + 0.529i)18-s + ⋯
L(s)  = 1  + (−0.241 − 0.970i)2-s + (−0.374 − 0.927i)3-s + (−0.882 + 0.469i)4-s + (0.961 − 0.275i)5-s + (−0.809 + 0.587i)6-s + (0.0348 − 0.999i)7-s + (0.669 + 0.743i)8-s + (−0.719 + 0.694i)9-s + (−0.5 − 0.866i)10-s + (0.766 + 0.642i)12-s + (0.438 − 0.898i)13-s + (−0.978 + 0.207i)14-s + (−0.615 − 0.788i)15-s + (0.559 − 0.829i)16-s + (0.438 + 0.898i)17-s + (0.848 + 0.529i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.960 + 0.276i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.960 + 0.276i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1334209588 - 0.9451772466i\)
\(L(\frac12)\) \(\approx\) \(-0.1334209588 - 0.9451772466i\)
\(L(1)\) \(\approx\) \(0.4802857261 - 0.7228558621i\)
\(L(1)\) \(\approx\) \(0.4802857261 - 0.7228558621i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.241 - 0.970i)T \)
3 \( 1 + (-0.374 - 0.927i)T \)
5 \( 1 + (0.961 - 0.275i)T \)
7 \( 1 + (0.0348 - 0.999i)T \)
13 \( 1 + (0.438 - 0.898i)T \)
17 \( 1 + (0.438 + 0.898i)T \)
19 \( 1 + (-0.374 - 0.927i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.990 + 0.139i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 + (0.961 + 0.275i)T \)
59 \( 1 + (-0.615 - 0.788i)T \)
61 \( 1 + (-0.719 - 0.694i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.241 + 0.970i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.559 + 0.829i)T \)
83 \( 1 + (0.438 + 0.898i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (-0.104 + 0.994i)T \)
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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

−25.01492024000078421906554855145, −24.00991436589521454393523041699, −22.94320866252319385935704505287, −22.300918444503516697560839096975, −21.46315827351279699692033000987, −20.895279514362794187149981443551, −19.29494046635846587504219035410, −18.24167736917694889911641917157, −17.87236825316929876587968110066, −16.583418705028395234082969782128, −16.28481791542293036882859880041, −15.13014976341904299196723764278, −14.48025242574200072442377391642, −13.68925562206017089735440383743, −12.35201350191235242224852077655, −11.173894041549734100249092421617, −10.104952229169420181784301252234, −9.29557889010824040126083049921, −8.84356880075810616412673534385, −7.34394557822754872463137907441, −5.99565050925022797517206882340, −5.75587372357270798845326976690, −4.6779093690706307489938175614, −3.39972885092405780912429907081, −1.75328747437046817746228944859, 0.663734270842529475993608142495, 1.63667783806934765747019523847, 2.66096668077841762088293691674, 4.05629218566264449920643453512, 5.30770159813239107110733098096, 6.276069669071474923305489895191, 7.58046106837918292377354862460, 8.40453758480567571452288949485, 9.561530151698798546121260606210, 10.648976790870538873703358105688, 11.048231243541138133542996090769, 12.54770823339570127578897798705, 12.97212958964268740121018325503, 13.683011963609068173246784887061, 14.56415599562546860745777373234, 16.54549986000753325717781926367, 17.17962351581207244123781690025, 17.812165480981216202739796468657, 18.54612879227778120643709023791, 19.61879798770898301844969391616, 20.274617727349708092989388339988, 21.099084269671838202127733668742, 22.15382128911695437772607273684, 22.84860167376956862798786731401, 23.76941226238888905219864618751

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