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Properties

Label 2-2271-2271.2057-c0-0-0
Degree $2$
Conductor $2271$
Sign $0.958 - 0.284i$
Analytic cond. $1.13337$
Root an. cond. $1.06460$
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  + (−0.5 + 0.866i)3-s + (0.542 − 0.840i)4-s + (−0.241 − 0.373i)7-s + (−0.499 − 0.866i)9-s + (0.456 + 0.889i)12-s + (0.0468 + 1.87i)13-s + (−0.411 − 0.911i)16-s + (0.655 + 0.496i)19-s + (0.444 − 0.0221i)21-s + (0.921 − 0.388i)25-s + 0.999·27-s − 0.445·28-s + (0.996 + 1.38i)31-s + (−0.998 − 0.0498i)36-s + (0.655 − 1.45i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.542 − 0.840i)4-s + (−0.241 − 0.373i)7-s + (−0.499 − 0.866i)9-s + (0.456 + 0.889i)12-s + (0.0468 + 1.87i)13-s + (−0.411 − 0.911i)16-s + (0.655 + 0.496i)19-s + (0.444 − 0.0221i)21-s + (0.921 − 0.388i)25-s + 0.999·27-s − 0.445·28-s + (0.996 + 1.38i)31-s + (−0.998 − 0.0498i)36-s + (0.655 − 1.45i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2271 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2271 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2271\)    =    \(3 \cdot 757\)
Sign: $0.958 - 0.284i$
Analytic conductor: \(1.13337\)
Root analytic conductor: \(1.06460\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2271} (2057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2271,\ (\ :0),\ 0.958 - 0.284i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.124000208\)
\(L(\frac12)\) \(\approx\) \(1.124000208\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
757 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.542 + 0.840i)T^{2} \)
5 \( 1 + (-0.921 + 0.388i)T^{2} \)
7 \( 1 + (0.241 + 0.373i)T + (-0.411 + 0.911i)T^{2} \)
11 \( 1 + (0.797 - 0.603i)T^{2} \)
13 \( 1 + (-0.0468 - 1.87i)T + (-0.998 + 0.0498i)T^{2} \)
17 \( 1 + (0.0249 - 0.999i)T^{2} \)
19 \( 1 + (-0.655 - 0.496i)T + (0.270 + 0.962i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.996 - 1.38i)T + (-0.318 + 0.947i)T^{2} \)
37 \( 1 + (-0.655 + 1.45i)T + (-0.661 - 0.749i)T^{2} \)
41 \( 1 + (-0.980 - 0.198i)T^{2} \)
43 \( 1 + (-1.82 - 0.463i)T + (0.878 + 0.478i)T^{2} \)
47 \( 1 + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (0.222 - 0.974i)T^{2} \)
61 \( 1 + (-0.137 + 1.83i)T + (-0.988 - 0.149i)T^{2} \)
67 \( 1 + (1.48 + 1.24i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (-0.270 + 0.962i)T^{2} \)
73 \( 1 + (-1.26 - 1.06i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (0.623 + 0.781i)T + (-0.222 + 0.974i)T^{2} \)
83 \( 1 + (0.939 - 0.342i)T^{2} \)
89 \( 1 + (0.0249 + 0.999i)T^{2} \)
97 \( 1 + (0.795 - 1.23i)T + (-0.411 - 0.911i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.380417519373864066342740573856, −8.846893729998672549934904750267, −7.49510573478425127427617663760, −6.61618458226644621476416790404, −6.24744592333108843839384209611, −5.21845873722608742996979466320, −4.54053931467872181768675944724, −3.67809434613618834276163166181, −2.46743171095230149779420553700, −1.13107748526595795190935201544, 1.05422966083167369418675461699, 2.74294897809255072594728181034, 2.85621861130775651085485440512, 4.36973062031525973470188949937, 5.53311536870817842016342013607, 6.03590164573004302637510135778, 6.97387073002097469229172192187, 7.62377076976478655807318550630, 8.140792373819169165586658473781, 8.923747356835922805664843829815

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