Properties

Label 1-1441-1441.588-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.999 + 0.0394i$
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  + (−0.943 + 0.331i)2-s + (−0.989 − 0.144i)3-s + (0.779 − 0.626i)4-s + (−0.168 − 0.985i)5-s + (0.981 − 0.192i)6-s + (0.120 + 0.992i)7-s + (−0.527 + 0.849i)8-s + (0.958 + 0.285i)9-s + (0.485 + 0.873i)10-s + (−0.861 + 0.506i)12-s + (0.120 + 0.992i)13-s + (−0.443 − 0.896i)14-s + (0.0241 + 0.999i)15-s + (0.215 − 0.976i)16-s + (−0.861 − 0.506i)17-s + (−0.998 + 0.0483i)18-s + ⋯
L(s)  = 1  + (−0.943 + 0.331i)2-s + (−0.989 − 0.144i)3-s + (0.779 − 0.626i)4-s + (−0.168 − 0.985i)5-s + (0.981 − 0.192i)6-s + (0.120 + 0.992i)7-s + (−0.527 + 0.849i)8-s + (0.958 + 0.285i)9-s + (0.485 + 0.873i)10-s + (−0.861 + 0.506i)12-s + (0.120 + 0.992i)13-s + (−0.443 − 0.896i)14-s + (0.0241 + 0.999i)15-s + (0.215 − 0.976i)16-s + (−0.861 − 0.506i)17-s + (−0.998 + 0.0483i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.999 + 0.0394i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (588, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.999 + 0.0394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6217546375 + 0.01226750510i\)
\(L(\frac12)\) \(\approx\) \(0.6217546375 + 0.01226750510i\)
\(L(1)\) \(\approx\) \(0.5354410893 + 0.01523179976i\)
\(L(1)\) \(\approx\) \(0.5354410893 + 0.01523179976i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.943 - 0.331i)T \)
3 \( 1 + (0.989 + 0.144i)T \)
5 \( 1 + (0.168 + 0.985i)T \)
7 \( 1 + (-0.120 - 0.992i)T \)
13 \( 1 + (-0.120 - 0.992i)T \)
17 \( 1 + (0.861 + 0.506i)T \)
19 \( 1 + (-0.399 + 0.916i)T \)
23 \( 1 + (-0.926 - 0.377i)T \)
29 \( 1 + (-0.568 - 0.822i)T \)
31 \( 1 + (0.0724 + 0.997i)T \)
37 \( 1 + (0.354 + 0.935i)T \)
41 \( 1 + (-0.995 - 0.0965i)T \)
43 \( 1 + (0.168 + 0.985i)T \)
47 \( 1 + (-0.958 - 0.285i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.861 + 0.506i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.168 - 0.985i)T \)
71 \( 1 + (-0.644 - 0.764i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.443 - 0.896i)T \)
83 \( 1 + (-0.779 - 0.626i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.906 - 0.421i)T \)
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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

−20.669653320053415217034020871023, −19.85279625916898032122366990376, −19.09044405703179912038374774751, −18.325759941198712986834659755981, −17.669746538425051433941190714351, −17.23991984099454402249615570985, −16.397712776775659290624567048555, −15.60028053556104276008717102267, −14.99892184537183598009740981961, −13.79028444319104537299799304821, −12.82250415131153142527119096028, −12.05990215223939512539660159237, −11.15206595817289377414632558802, −10.59461377468762476104012939983, −10.338253292751588362720432981168, −9.363504334514231559058116893528, −8.037696520730191622279179750780, −7.50905282441128327962048622990, −6.59377025132584515835536512397, −6.14404592237379649259074677382, −4.72917692392930299063812397993, −3.733056063939663008569149600358, −2.97906387629928451234785179188, −1.645750908680028649868724810225, −0.65413598117217902262120909482, 0.65605221444624839280589938193, 1.633695856447018226112002354702, 2.525066357644912916731370204501, 4.307797731402416233396623269457, 5.16708706904305068366235338914, 5.69177681799784130435435823186, 6.731733964434861654363221350544, 7.31318768693461715683680483239, 8.45262793662783838299391217485, 9.17460283887351264146390214879, 9.55156052736396147850955221801, 10.978104078022115020048710282959, 11.38013166163911435418234383069, 12.08830734331636688992305399040, 12.85078812244694594417886946118, 13.87036493866225988649730705484, 15.11164898702995354681238780556, 15.93869562467866993409902645342, 16.07183376342304305979272427722, 17.16121198120326085751054494534, 17.55547145103615954287938595139, 18.39032909477127596801924014158, 19.02507902916521787311305675935, 19.75046595952386278618015702896, 20.71730695946704899290554042735

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