Properties

Degree $2$
Conductor $252$
Sign $0.266 - 0.963i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)5-s + (0.5 + 2.59i)7-s + (1 + 1.73i)11-s − 3·13-s + (4 + 6.92i)17-s + (0.5 − 0.866i)19-s + (4 − 6.92i)23-s + (0.500 + 0.866i)25-s − 4·29-s + (−1.5 − 2.59i)31-s + (−5 − 1.73i)35-s + (0.5 − 0.866i)37-s − 6·41-s + 11·43-s + (3 − 5.19i)47-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + (0.188 + 0.981i)7-s + (0.301 + 0.522i)11-s − 0.832·13-s + (0.970 + 1.68i)17-s + (0.114 − 0.198i)19-s + (0.834 − 1.44i)23-s + (0.100 + 0.173i)25-s − 0.742·29-s + (−0.269 − 0.466i)31-s + (−0.845 − 0.292i)35-s + (0.0821 − 0.142i)37-s − 0.937·41-s + 1.67·43-s + (0.437 − 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.266 - 0.963i$
Motivic weight: \(1\)
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.894481 + 0.680481i\)
\(L(\frac12)\) \(\approx\) \(0.894481 + 0.680481i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + (-4 - 6.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−12.31452891717860794310765759901, −11.28791527001990603061071035990, −10.42924890499627037924724190065, −9.372159469906026931154183920467, −8.299391711674754658549173330954, −7.30990981320539886578536057919, −6.25554651021895590681373542408, −5.04613988998220734418384382401, −3.60203331204938496924430643433, −2.23241068812336342992105072826, 0.965795568859102392831399663720, 3.25553039821128219080739947707, 4.54400292622656144790669268154, 5.48305515935114243213300889725, 7.19961418344622387872940235902, 7.71444404342075596320824127988, 9.042130284714622882543932201253, 9.812583697983873621121745053008, 11.04531449514874741828729448281, 11.83503703506292502175025376957

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