Properties

Label 1-1441-1441.725-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.00991 + 0.999i$
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  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)6-s + (0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.309 + 0.951i)10-s + (−0.809 + 0.587i)12-s + (−0.309 + 0.951i)13-s + (−0.309 − 0.951i)14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + 18-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)6-s + (0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.309 + 0.951i)10-s + (−0.809 + 0.587i)12-s + (−0.309 + 0.951i)13-s + (−0.309 − 0.951i)14-s + (0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00991 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00991 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.00991 + 0.999i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (725, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.00991 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6376163955 + 0.6439684798i\)
\(L(\frac12)\) \(\approx\) \(0.6376163955 + 0.6439684798i\)
\(L(1)\) \(\approx\) \(0.7324133858 + 0.1618182535i\)
\(L(1)\) \(\approx\) \(0.7324133858 + 0.1618182535i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 - T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−20.09965664227761587593027862587, −19.71718309105473355885985281283, −18.92472804673151525789474058406, −18.18328431615350547781076223931, −17.74039419916142472951403940770, −16.98308324495111575705339176087, −16.0324054354005307918465907112, −15.15557232470950957807115424646, −14.41723787836307839808712353605, −14.17866721927458039550584604128, −12.871377091479532782910602552982, −12.056391733215122985408496683815, −11.08067990372985607651175146979, −10.663102812814610113698715956196, −9.59195230266282311953612353906, −8.44411331395764010137293874329, −7.85550675976852486117077070901, −7.55045653926556580201252118201, −6.6634086673351387183830908483, −5.844905873102684164628994546757, −4.77509651432944974291142701503, −3.51167500929862941615904560789, −2.51083621374953457295095315532, −1.424421281759182472829784917064, −0.51355757356015184934153822588, 1.11469193418776580264224322235, 2.26824495968072941417046667778, 3.13710655394231932907222332354, 4.13116547449861679195920831723, 4.70772808458148217598730142743, 5.69764195173802702071245511857, 7.35038839726831623909952253930, 7.886263704685160067058685928182, 8.63363427788173099821021306507, 9.476317962306504047473335046036, 9.73245833977707195248603867523, 11.16845234688487387625043958900, 11.58664755764028760383272464347, 11.9514452668980825375588208747, 13.2396527466626222435111285265, 14.15646773639695575898393211607, 15.1095018930214363764693897614, 15.833067090188424663596474465095, 16.36557031321468230439783879808, 17.08525679866973975430381046444, 17.94837539081807684275434879451, 18.86580716764191291263829988018, 19.45562228768486987178335482070, 20.27250907702203442655647155072, 20.82442782354962983123677979657

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