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Properties

Label 1-1441-1441.961-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.212 + 0.977i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (−0.607 − 0.794i)2-s + (−0.168 − 0.985i)3-s + (−0.262 + 0.964i)4-s + (−0.443 + 0.896i)5-s + (−0.681 + 0.732i)6-s + (0.120 + 0.992i)7-s + (0.926 − 0.377i)8-s + (−0.943 + 0.331i)9-s + (0.981 − 0.192i)10-s + (0.995 + 0.0965i)12-s + (0.120 + 0.992i)13-s + (0.715 − 0.698i)14-s + (0.958 + 0.285i)15-s + (−0.861 − 0.506i)16-s + (0.995 − 0.0965i)17-s + (0.836 + 0.548i)18-s + ⋯
L(s)  = 1  + (−0.607 − 0.794i)2-s + (−0.168 − 0.985i)3-s + (−0.262 + 0.964i)4-s + (−0.443 + 0.896i)5-s + (−0.681 + 0.732i)6-s + (0.120 + 0.992i)7-s + (0.926 − 0.377i)8-s + (−0.943 + 0.331i)9-s + (0.981 − 0.192i)10-s + (0.995 + 0.0965i)12-s + (0.120 + 0.992i)13-s + (0.715 − 0.698i)14-s + (0.958 + 0.285i)15-s + (−0.861 − 0.506i)16-s + (0.995 − 0.0965i)17-s + (0.836 + 0.548i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.212 + 0.977i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.212 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5208891805 + 0.4199709693i\)
\(L(\frac12)\) \(\approx\) \(0.5208891805 + 0.4199709693i\)
\(L(1)\) \(\approx\) \(0.6498116311 - 0.06187429007i\)
\(L(1)\) \(\approx\) \(0.6498116311 - 0.06187429007i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.607 - 0.794i)T \)
3 \( 1 + (-0.168 - 0.985i)T \)
5 \( 1 + (-0.443 + 0.896i)T \)
7 \( 1 + (0.120 + 0.992i)T \)
13 \( 1 + (0.120 + 0.992i)T \)
17 \( 1 + (0.995 - 0.0965i)T \)
19 \( 1 + (0.215 + 0.976i)T \)
23 \( 1 + (-0.0724 + 0.997i)T \)
29 \( 1 + (0.568 + 0.822i)T \)
31 \( 1 + (0.644 + 0.764i)T \)
37 \( 1 + (-0.354 - 0.935i)T \)
41 \( 1 + (0.399 - 0.916i)T \)
43 \( 1 + (-0.443 + 0.896i)T \)
47 \( 1 + (-0.943 + 0.331i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.995 - 0.0965i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.443 - 0.896i)T \)
71 \( 1 + (-0.527 + 0.849i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.715 + 0.698i)T \)
83 \( 1 + (-0.262 - 0.964i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.485 - 0.873i)T \)
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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

−20.47280024656410926023330520138, −19.89332136007224341333668881419, −19.18165310961559388041192585759, −17.95420184227311062405052925217, −17.28019893798351660281404950248, −16.71185542328728816819643917687, −16.21735173880738253743973495635, −15.36630996047348556658985261076, −14.9059373049303553939132870414, −13.852010394870012949556645921200, −13.19130982884362721384388552242, −11.93545079636721938207092804895, −11.11907465363222279510172213667, −10.16416789530817226902334182799, −9.896835583759328223539210743496, −8.75288464548501311640099361901, −8.19983938246271838821723231712, −7.482437770318270146378249596520, −6.33940507802742239772677315823, −5.44975946432973577489280949899, −4.69451909271031501750368410736, −4.148136892787481276146618588719, −2.948748599863961984037837851743, −1.1254396139932823381373333765, −0.386571446181035253640896146237, 1.33993201684684781495060331785, 2.039053056095317866584785128589, 2.99291934303176210000130609723, 3.62501313732291745421876509958, 5.058269401770809685240043596712, 6.120697451682547923561091362561, 6.97821568882017227494246850551, 7.73209094222401979770864948081, 8.37672197629173035319347751983, 9.24965245630576484583949944324, 10.1989490654233341991167847349, 11.09864577036114612202063631998, 11.88865419928510475293749864395, 12.04257159685044330574493010197, 12.985814108046806695896302519761, 14.15117061420966267341411993143, 14.431195940834408022278236343209, 15.85171983059075433638459528151, 16.46312917829129571004503419116, 17.57050877272742038105482709114, 18.15284777958071206346251330885, 18.65350028289087266174833683780, 19.34586820164464502173835795901, 19.61033458463570224643697434256, 20.99810609578101758803587379576

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