↑

Properties

Label 2-252-84.83-c0-0-1
Degree $2$
Conductor $252$
Sign $0.577 + 0.816i$
Analytic cond. $0.125764$
Root an. cond. $0.354632$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s − i·7-s + (0.707 − 0.707i)8-s + 1.41·11-s + (−0.707 + 0.707i)14-s − 1.00·16-s + (−1.00 − 1.00i)22-s − 1.41·23-s − 25-s + 1.00·28-s + 1.41i·29-s + (0.707 + 0.707i)32-s + 2i·43-s + 1.41i·44-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s − i·7-s + (0.707 − 0.707i)8-s + 1.41·11-s + (−0.707 + 0.707i)14-s − 1.00·16-s + (−1.00 − 1.00i)22-s − 1.41·23-s − 25-s + 1.00·28-s + 1.41i·29-s + (0.707 + 0.707i)32-s + 2i·43-s + 1.41i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(0.125764\)
Root analytic conductor: \(0.354632\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :0),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5560245805\)
\(L(\frac12)\) \(\approx\) \(0.5560245805\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 2iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.87637549358460057796506369175, −11.19078969887044824848430645090, −10.14462672566241191937119273317, −9.464625684446966374756723274292, −8.359760427737248794567555870120, −7.37636580278176534769287781777, −6.38076241734211654654451308769, −4.37519698725907040728084771910, −3.48611782702902368647548111112, −1.56459043903912568487343958848, 2.00353902930501664458134600861, 4.15660480590871533776607540093, 5.70558301503066036626691818457, 6.34382723285042271943705451271, 7.60362404025781515972335137591, 8.616183523062858570048153568749, 9.359459523501214936299721487488, 10.19447724769850441126960524225, 11.57002336867066918285115562707, 12.08345095106201476585867173593

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