Properties

Label 1-1441-1441.1087-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.447 + 0.894i$
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  + (−0.262 + 0.964i)2-s + (−0.607 − 0.794i)3-s + (−0.861 − 0.506i)4-s + (0.0241 + 0.999i)5-s + (0.926 − 0.377i)6-s + (−0.527 − 0.849i)7-s + (0.715 − 0.698i)8-s + (−0.262 + 0.964i)9-s + (−0.970 − 0.239i)10-s + (0.120 + 0.992i)12-s + (0.926 + 0.377i)13-s + (0.958 − 0.285i)14-s + (0.779 − 0.626i)15-s + (0.485 + 0.873i)16-s + (0.485 + 0.873i)17-s + (−0.861 − 0.506i)18-s + ⋯
L(s)  = 1  + (−0.262 + 0.964i)2-s + (−0.607 − 0.794i)3-s + (−0.861 − 0.506i)4-s + (0.0241 + 0.999i)5-s + (0.926 − 0.377i)6-s + (−0.527 − 0.849i)7-s + (0.715 − 0.698i)8-s + (−0.262 + 0.964i)9-s + (−0.970 − 0.239i)10-s + (0.120 + 0.992i)12-s + (0.926 + 0.377i)13-s + (0.958 − 0.285i)14-s + (0.779 − 0.626i)15-s + (0.485 + 0.873i)16-s + (0.485 + 0.873i)17-s + (−0.861 − 0.506i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7582087521 + 0.4682578978i\)
\(L(\frac12)\) \(\approx\) \(0.7582087521 + 0.4682578978i\)
\(L(1)\) \(\approx\) \(0.6857814932 + 0.2315656471i\)
\(L(1)\) \(\approx\) \(0.6857814932 + 0.2315656471i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.262 + 0.964i)T \)
3 \( 1 + (-0.607 - 0.794i)T \)
5 \( 1 + (0.0241 + 0.999i)T \)
7 \( 1 + (-0.527 - 0.849i)T \)
13 \( 1 + (0.926 + 0.377i)T \)
17 \( 1 + (0.485 + 0.873i)T \)
19 \( 1 + (0.981 + 0.192i)T \)
23 \( 1 + (0.885 - 0.464i)T \)
29 \( 1 + (-0.998 - 0.0483i)T \)
31 \( 1 + (-0.443 - 0.896i)T \)
37 \( 1 + (-0.861 - 0.506i)T \)
41 \( 1 + (0.981 + 0.192i)T \)
43 \( 1 + (0.568 - 0.822i)T \)
47 \( 1 + (-0.998 + 0.0483i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.981 - 0.192i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (0.568 + 0.822i)T \)
71 \( 1 + (-0.989 + 0.144i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.943 + 0.331i)T \)
83 \( 1 + (0.995 + 0.0965i)T \)
89 \( 1 + T \)
97 \( 1 + (0.926 + 0.377i)T \)
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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

−20.74310318240939716195949219504, −20.062245477525109838010917661481, −19.160757498745889476035940417881, −18.259383943958916675939898598706, −17.73131998633662896157078498029, −16.82682025407382496743554629992, −16.09333370779549048962750324159, −15.69260077447436516966279933956, −14.47244824210505390758430322329, −13.39866487217174023199423221664, −12.74933572098350345715740753624, −12.01114427794061344877245576038, −11.44575245226233923387224741866, −10.65188491373326462769462992502, −9.55050801268287886248976283170, −9.30245371408757378807820537334, −8.615159036442726922395431079887, −7.502797916246405843981604637, −6.01919331555728382204961090176, −5.25329729441028932951154979155, −4.78015068642371163099537809709, −3.496656524098923385969616105680, −3.10640851117743846160178284882, −1.57654402850289679827109707354, −0.63115155461990347115692975034, 0.79085837612466556442850691499, 1.81475028487436317420794637248, 3.347940124572865251708312035086, 4.12385423192513561185690580677, 5.48466250543039904880709757729, 6.08176011983451078648326352995, 6.80540533763514394751376314863, 7.366536826151484329932878889162, 8.00433883283692143734633127022, 9.15994870675668229556750467912, 10.14316064376618341225316465121, 10.77910390047773533235977413373, 11.47312382873489739241399668120, 12.833487132611455380813964185616, 13.29278486329512410961519392085, 14.14337404943361452445615784610, 14.644107530251647163303803708659, 15.81381083683756122681417415781, 16.40155998713766472498758320667, 17.16557521267568816291244531390, 17.73807564281602110265080290052, 18.67030511926881776026296292298, 18.91267800726498936444418989480, 19.68348017327841002743180982527, 20.87383722566690803706758685044

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