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Properties

Label 2-252-84.23-c1-0-3
Degree $2$
Conductor $252$
Sign $0.982 + 0.185i$
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  + (−0.171 − 1.40i)2-s + (−1.94 + 0.480i)4-s + (3.35 + 1.93i)5-s + (−1.03 + 2.43i)7-s + (1.00 + 2.64i)8-s + (2.14 − 5.03i)10-s + (1.73 + 3.00i)11-s − 0.296·13-s + (3.59 + 1.03i)14-s + (3.53 − 1.86i)16-s + (1.35 − 0.783i)17-s + (−6.12 − 3.53i)19-s + (−7.43 − 2.14i)20-s + (3.92 − 2.95i)22-s + (2.71 − 4.70i)23-s + ⋯
L(s)  = 1  + (−0.121 − 0.992i)2-s + (−0.970 + 0.240i)4-s + (1.49 + 0.865i)5-s + (−0.390 + 0.920i)7-s + (0.356 + 0.934i)8-s + (0.677 − 1.59i)10-s + (0.523 + 0.905i)11-s − 0.0822·13-s + (0.961 + 0.275i)14-s + (0.884 − 0.466i)16-s + (0.329 − 0.190i)17-s + (−1.40 − 0.811i)19-s + (−1.66 − 0.479i)20-s + (0.835 − 0.628i)22-s + (0.566 − 0.981i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.982 + 0.185i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.982 + 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28940 - 0.120700i\)
\(L(\frac12)\) \(\approx\) \(1.28940 - 0.120700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.171 + 1.40i)T \)
3 \( 1 \)
7 \( 1 + (1.03 - 2.43i)T \)
good5 \( 1 + (-3.35 - 1.93i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.73 - 3.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.296T + 13T^{2} \)
17 \( 1 + (-1.35 + 0.783i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.12 + 3.53i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.71 + 4.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.85iT - 29T^{2} \)
31 \( 1 + (-2.43 + 1.40i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.25 - 2.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.55iT - 41T^{2} \)
43 \( 1 - 0.682iT - 43T^{2} \)
47 \( 1 + (-1.18 + 2.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.540 - 0.311i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.42 - 7.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.33 - 2.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.19 + 5.30i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.539T + 71T^{2} \)
73 \( 1 + (3.69 + 6.40i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.33 + 3.08i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.15T + 83T^{2} \)
89 \( 1 + (10.1 + 5.85i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.84T + 97T^{2} \)
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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

−12.01418743496038751927813328409, −10.93861921442971343118601028401, −10.02678967148110006587272352716, −9.494480862830005284843937761877, −8.579445348521417858933516550958, −6.82668874774211899318840012697, −5.90640887613808797508700040978, −4.59231541718998900835766374366, −2.79398688832245449276070541859, −2.07794142877401736735440941378, 1.26718763975723035293867143810, 3.81148109528509678669805730179, 5.17220036799458234385263131452, 6.04044845222676259769025652810, 6.87253109579945274513849151864, 8.299356399368628738512737328773, 9.082732668435249630660346213610, 9.907676398451166081915903718699, 10.72360524264404902479906514058, 12.60759124602760950129005404248

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