Properties

Label 2-252-28.27-c1-0-12
Degree $2$
Conductor $252$
Sign $0.608 + 0.793i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.780 + 1.17i)2-s + (−0.780 − 1.84i)4-s − 1.69i·5-s + (−2.56 + 0.662i)7-s + (2.78 + 0.516i)8-s + (2 + 1.32i)10-s − 3.02i·11-s − 6.04i·13-s + (1.21 − 3.53i)14-s + (−2.78 + 2.87i)16-s − 4.34i·17-s + 1.12·19-s + (−3.12 + 1.32i)20-s + (3.56 + 2.35i)22-s + 3.02i·23-s + ⋯
L(s)  = 1  + (−0.552 + 0.833i)2-s + (−0.390 − 0.920i)4-s − 0.758i·5-s + (−0.968 + 0.250i)7-s + (0.983 + 0.182i)8-s + (0.632 + 0.418i)10-s − 0.910i·11-s − 1.67i·13-s + (0.325 − 0.945i)14-s + (−0.695 + 0.718i)16-s − 1.05i·17-s + 0.257·19-s + (−0.698 + 0.296i)20-s + (0.759 + 0.502i)22-s + 0.629i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.608 + 0.793i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.623632 - 0.307729i\)
\(L(\frac12)\) \(\approx\) \(0.623632 - 0.307729i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.780 - 1.17i)T \)
3 \( 1 \)
7 \( 1 + (2.56 - 0.662i)T \)
good5 \( 1 + 1.69iT - 5T^{2} \)
11 \( 1 + 3.02iT - 11T^{2} \)
13 \( 1 + 6.04iT - 13T^{2} \)
17 \( 1 + 4.34iT - 17T^{2} \)
19 \( 1 - 1.12T + 19T^{2} \)
23 \( 1 - 3.02iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 7.12T + 37T^{2} \)
41 \( 1 - 7.73iT - 41T^{2} \)
43 \( 1 + 8.10iT - 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 9.43iT - 61T^{2} \)
67 \( 1 - 2.06iT - 67T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 - 3.39iT - 73T^{2} \)
79 \( 1 - 4.71iT - 79T^{2} \)
83 \( 1 - 6.24T + 83T^{2} \)
89 \( 1 + 7.73iT - 89T^{2} \)
97 \( 1 - 8.68iT - 97T^{2} \)
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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

−11.95591648484610419118707777558, −10.63500972691350806749014516502, −9.767422053145232604109193081730, −8.884323767157839161638134140045, −8.112271176030091713646032641211, −6.96874588769461866320379606334, −5.78057953225196759515243928198, −5.08843144426301272963333119371, −3.20508308914145837452820012099, −0.66884033543823757728129316142, 2.00998914018690913983320959374, 3.39538406667254509395169331491, 4.46711452391305228634967994700, 6.57083924130414393763449692854, 7.14431026890047472265958664964, 8.561676929671817800086832661022, 9.559191489201107256203160067156, 10.24009029761968135095983801344, 11.08954943750795433849530752053, 12.13043002636508401933059666395

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