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Properties

Label 1-1224-1224.1069-r0-0-0
Degree $1$
Conductor $1224$
Sign $-0.699 - 0.715i$
Analytic cond. $5.68423$
Root an. cond. $5.68423$
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  + (−0.258 − 0.965i)5-s + (0.258 − 0.965i)7-s + (0.258 − 0.965i)11-s + (−0.5 + 0.866i)13-s − i·19-s + (0.965 − 0.258i)23-s + (−0.866 + 0.5i)25-s + (0.965 + 0.258i)29-s + (−0.258 − 0.965i)31-s − 35-s + (0.707 − 0.707i)37-s + (−0.965 + 0.258i)41-s + (0.866 − 0.5i)43-s + (0.5 + 0.866i)47-s + (−0.866 − 0.5i)49-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)5-s + (0.258 − 0.965i)7-s + (0.258 − 0.965i)11-s + (−0.5 + 0.866i)13-s − i·19-s + (0.965 − 0.258i)23-s + (−0.866 + 0.5i)25-s + (0.965 + 0.258i)29-s + (−0.258 − 0.965i)31-s − 35-s + (0.707 − 0.707i)37-s + (−0.965 + 0.258i)41-s + (0.866 − 0.5i)43-s + (0.5 + 0.866i)47-s + (−0.866 − 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $-0.699 - 0.715i$
Analytic conductor: \(5.68423\)
Root analytic conductor: \(5.68423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (1069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1224,\ (0:\ ),\ -0.699 - 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4832854174 - 1.148347558i\)
\(L(\frac12)\) \(\approx\) \(0.4832854174 - 1.148347558i\)
\(L(1)\) \(\approx\) \(0.9106009135 - 0.4395117233i\)
\(L(1)\) \(\approx\) \(0.9106009135 - 0.4395117233i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
11 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 - iT \)
23 \( 1 + (0.965 - 0.258i)T \)
29 \( 1 + (0.965 + 0.258i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (-0.965 + 0.258i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (-0.258 + 0.965i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (-0.258 + 0.965i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.965 - 0.258i)T \)
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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

−21.62439759664750374748253417099, −20.62464201208002931012090084362, −19.831885849672186597706834208184, −19.04027080060929054664634861046, −18.35091710184468636793475981780, −17.742273401522212807841200449658, −16.97879872299273763419065050937, −15.73093045003347377323482082164, −15.17280264332350613750602945632, −14.68829597176584252508745851200, −13.8578039500000567003118437058, −12.58161570116990369125911071110, −12.16762220451039625422501209774, −11.293565350280625852783727864612, −10.36825425611475113812971336434, −9.77273599131149173274406028914, −8.71054575114560606116417349277, −7.83101645971876084879749493935, −7.10894314712155678834465261919, −6.2036381195623453550939567842, −5.32140287794569234450212996300, −4.40322338000134380841551163549, −3.19497372695891605947849849786, −2.571820311241397250156321039033, −1.4970145022468181541589816935, 0.52778616244968737665704721444, 1.37145343708912465741908009360, 2.69845856035656217726397474323, 3.90911031922596654808478470612, 4.52670798353455681888506374578, 5.329266839982814810867470580659, 6.509213367495459304487372461579, 7.300176344462451913723384764168, 8.1904694020997280884479768292, 8.98710545282199318433471672606, 9.63250918173369011055742574234, 10.87725347504950212013726636040, 11.36256067649038781288248102237, 12.283459387779615932594871166006, 13.18274659953382368797542684620, 13.78490500194897463450483120687, 14.54953111199769991804540032183, 15.59829949408301183735261651551, 16.435663393515351412000861269546, 16.92213696448747278079900706770, 17.51434992991150507067268019409, 18.703692787911931913108654111547, 19.52957219730154141012019783032, 19.95178120601307353032946076814, 20.912503776848573862710111472303

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