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Properties

Label 1-1441-1441.1082-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.889 + 0.456i$
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  + (−0.970 − 0.239i)2-s + (0.981 − 0.192i)3-s + (0.885 + 0.464i)4-s + (−0.681 + 0.732i)5-s + (−0.998 − 0.0483i)6-s + (0.779 − 0.626i)7-s + (−0.748 − 0.663i)8-s + (0.926 − 0.377i)9-s + (0.836 − 0.548i)10-s + (0.958 + 0.285i)12-s + (−0.262 + 0.964i)13-s + (−0.906 + 0.421i)14-s + (−0.527 + 0.849i)15-s + (0.568 + 0.822i)16-s + (−0.607 + 0.794i)17-s + (−0.989 + 0.144i)18-s + ⋯
L(s)  = 1  + (−0.970 − 0.239i)2-s + (0.981 − 0.192i)3-s + (0.885 + 0.464i)4-s + (−0.681 + 0.732i)5-s + (−0.998 − 0.0483i)6-s + (0.779 − 0.626i)7-s + (−0.748 − 0.663i)8-s + (0.926 − 0.377i)9-s + (0.836 − 0.548i)10-s + (0.958 + 0.285i)12-s + (−0.262 + 0.964i)13-s + (−0.906 + 0.421i)14-s + (−0.527 + 0.849i)15-s + (0.568 + 0.822i)16-s + (−0.607 + 0.794i)17-s + (−0.989 + 0.144i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.889 + 0.456i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1082, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.889 + 0.456i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.292310925 + 0.3119879099i\)
\(L(\frac12)\) \(\approx\) \(1.292310925 + 0.3119879099i\)
\(L(1)\) \(\approx\) \(0.9573260106 + 0.01673242593i\)
\(L(1)\) \(\approx\) \(0.9573260106 + 0.01673242593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.970 - 0.239i)T \)
3 \( 1 + (0.981 - 0.192i)T \)
5 \( 1 + (-0.681 + 0.732i)T \)
7 \( 1 + (0.779 - 0.626i)T \)
13 \( 1 + (-0.262 + 0.964i)T \)
17 \( 1 + (-0.607 + 0.794i)T \)
19 \( 1 + (-0.943 + 0.331i)T \)
23 \( 1 + (0.215 - 0.976i)T \)
29 \( 1 + (-0.527 - 0.849i)T \)
31 \( 1 + (0.995 + 0.0965i)T \)
37 \( 1 + (-0.168 + 0.985i)T \)
41 \( 1 + (0.0241 + 0.999i)T \)
43 \( 1 + (0.981 - 0.192i)T \)
47 \( 1 + (0.644 + 0.764i)T \)
53 \( 1 + T \)
59 \( 1 + (0.568 + 0.822i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.981 + 0.192i)T \)
71 \( 1 + (-0.861 + 0.506i)T \)
73 \( 1 + T \)
79 \( 1 + (0.981 - 0.192i)T \)
83 \( 1 + (-0.989 - 0.144i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.836 + 0.548i)T \)
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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

−20.54473175520955419141575604420, −19.79049109936923485207266244352, −19.30889112227695185627050083358, −18.486540056840666201230108942116, −17.68443190282337626262060181390, −17.00148319785835125321726653629, −15.83595224141070683244168428822, −15.54728329483544694637087317434, −14.94752839495542878499338520930, −14.077167165614004013252465646660, −12.95613003064830956412207209763, −12.203878538408371570651317842, −11.26602457717150612227599937464, −10.58997861962815021448003610172, −9.45663394934446252772124497096, −8.90995490137028824564450723617, −8.33848525580282332893888557766, −7.655798083275372810510863930520, −7.0110112240728866238146214016, −5.51554881635436538273281785724, −4.88587453234420857741372896861, −3.73780641302697551309161745204, −2.606463265489078215526898219, −1.90085317063960253414071273743, −0.707510988424524712532233349950, 1.05092715503358062495929180066, 2.13362778384023858786182308275, 2.6810639788462185700235052064, 4.048989249380037059561303896088, 4.22782439938424625571622269377, 6.414329113530454510198891944738, 6.88233201897769236723244568919, 7.770494640168891863841947247720, 8.2780389444368767469127598153, 8.9488772867118691339696166579, 10.08002731570627749615396109996, 10.60098499717063714247782170977, 11.43466812026420920834004085822, 12.16609449749961740764487643265, 13.15247256193198715218225642723, 14.13889271645659188173539477724, 14.894704672539933448476656386925, 15.29607109659577409163713929522, 16.367016984018876934833316010730, 17.156571745571837313396396775599, 17.936862738296797811887556754378, 18.834726318060834150412408823766, 19.16917160058921026691874555670, 19.82362683828771973436688665823, 20.6543681034714018211419570554

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