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Properties

Label 2-252-7.3-c0-0-0
Degree $2$
Conductor $252$
Sign $0.991 - 0.126i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)7-s − 1.73i·13-s + (−1.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + (−1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + 43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)67-s + (1.5 + 0.866i)73-s + (0.5 + 0.866i)79-s + (1.49 − 0.866i)91-s + (1.5 − 0.866i)103-s + (−0.5 + 0.866i)109-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s − 1.73i·13-s + (−1.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + (−1.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + 43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)67-s + (1.5 + 0.866i)73-s + (0.5 + 0.866i)79-s + (1.49 − 0.866i)91-s + (1.5 − 0.866i)103-s + (−0.5 + 0.866i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(0.125764\)
Root analytic conductor: \(0.354632\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :0),\ 0.991 - 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7533523450\)
\(L(\frac12)\) \(\approx\) \(0.7533523450\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.73iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−12.53215418346690045431137012082, −11.24298370669853202270486110399, −10.53267159925723775319435664971, −9.356016630938252112091714826295, −8.344641000529020091597113578714, −7.58550144724325126417600350942, −5.99967589066269309991929784482, −5.29873520295787314240505170371, −3.72761380043161358497732490050, −2.18384318450784974480618441631, 1.98126574448115951940535471276, 3.95066593784166164853909707177, 4.78770231110273723801441485478, 6.43820644479442424950699200760, 7.19442353313728619277730897997, 8.434738428211970939090069030582, 9.311494397191903195635700960849, 10.53020636809064090326126324008, 11.20793083446542881049647399051, 12.17999464888387279957474132642

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