Properties

Label 2-252-84.23-c1-0-1
Degree $2$
Conductor $252$
Sign $-0.0581 - 0.998i$
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  + (−0.783 + 1.17i)2-s + (−0.771 − 1.84i)4-s + (2.15 + 1.24i)5-s + (−2.64 + 0.0803i)7-s + (2.77 + 0.537i)8-s + (−3.15 + 1.56i)10-s + (2.30 + 3.99i)11-s + 5.22·13-s + (1.97 − 3.17i)14-s + (−2.80 + 2.84i)16-s + (−4.85 + 2.80i)17-s + (2.76 + 1.59i)19-s + (0.632 − 4.93i)20-s + (−6.50 − 0.414i)22-s + (0.359 − 0.622i)23-s + ⋯
L(s)  = 1  + (−0.554 + 0.832i)2-s + (−0.385 − 0.922i)4-s + (0.963 + 0.556i)5-s + (−0.999 + 0.0303i)7-s + (0.981 + 0.189i)8-s + (−0.997 + 0.493i)10-s + (0.694 + 1.20i)11-s + 1.44·13-s + (0.528 − 0.848i)14-s + (−0.702 + 0.712i)16-s + (−1.17 + 0.680i)17-s + (0.634 + 0.366i)19-s + (0.141 − 1.10i)20-s + (−1.38 − 0.0884i)22-s + (0.0749 − 0.129i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.0581 - 0.998i$
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: \((2,\ 252,\ (\ :1/2),\ -0.0581 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692882 + 0.734453i\)
\(L(\frac12)\) \(\approx\) \(0.692882 + 0.734453i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.783 - 1.17i)T \)
3 \( 1 \)
7 \( 1 + (2.64 - 0.0803i)T \)
good5 \( 1 + (-2.15 - 1.24i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.30 - 3.99i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.22T + 13T^{2} \)
17 \( 1 + (4.85 - 2.80i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.76 - 1.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.359 + 0.622i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.53iT - 29T^{2} \)
31 \( 1 + (1.01 - 0.588i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.35 - 2.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.83iT - 41T^{2} \)
43 \( 1 + 11.1iT - 43T^{2} \)
47 \( 1 + (-2.70 + 4.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.79 + 1.03i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.05 + 3.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.505 + 0.874i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.9 + 6.32i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.31T + 71T^{2} \)
73 \( 1 + (4.81 + 8.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.65 - 4.41i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + (7.38 + 4.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−12.47692785272318499461512751344, −10.90272862419467672620371657953, −10.16023047478964381020288928328, −9.364809821918978281295785333093, −8.583152492978253570499818871690, −6.94819210068089792362374274664, −6.53761574477756620983998327342, −5.56112448160855926807115456678, −3.89413461806880510029460608921, −1.85625528372577994185536773356, 1.08733411789743035261470119319, 2.84370082502259892656992589552, 4.05246936837095291548060850891, 5.74821560693746682273035032569, 6.72697787396311544431387200610, 8.359634515668060253460492173844, 9.208460316508885570092305645958, 9.612801560459858021388264526494, 10.95454079162858639936532998141, 11.52752492294150915463422882442

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