Properties

Label 2-252-12.11-c7-0-37
Degree $2$
Conductor $252$
Sign $-0.890 - 0.455i$
Analytic cond. $78.7210$
Root an. cond. $8.87248$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.67 + 5.85i)2-s + (59.3 + 113. i)4-s + 181. i·5-s + 343i·7-s + (−89.4 + 1.44e3i)8-s + (−1.06e3 + 1.75e3i)10-s + 2.43e3·11-s + 9.05e3·13-s + (−2.00e3 + 3.32e3i)14-s + (−9.33e3 + 1.34e4i)16-s + 1.32e4i·17-s + 1.11e4i·19-s + (−2.05e4 + 1.07e4i)20-s + (2.35e4 + 1.42e4i)22-s + 3.40e4·23-s + ⋯
L(s)  = 1  + (0.855 + 0.517i)2-s + (0.463 + 0.885i)4-s + 0.647i·5-s + 0.377i·7-s + (−0.0617 + 0.998i)8-s + (−0.335 + 0.554i)10-s + 0.551·11-s + 1.14·13-s + (−0.195 + 0.323i)14-s + (−0.569 + 0.821i)16-s + 0.652i·17-s + 0.373i·19-s + (−0.573 + 0.300i)20-s + (0.471 + 0.285i)22-s + 0.582·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.890 - 0.455i$
Analytic conductor: \(78.7210\)
Root analytic conductor: \(8.87248\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :7/2),\ -0.890 - 0.455i)\)

Particular Values

\(L(4)\) \(\approx\) \(3.757345821\)
\(L(\frac12)\) \(\approx\) \(3.757345821\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.67 - 5.85i)T \)
3 \( 1 \)
7 \( 1 - 343iT \)
good5 \( 1 - 181. iT - 7.81e4T^{2} \)
11 \( 1 - 2.43e3T + 1.94e7T^{2} \)
13 \( 1 - 9.05e3T + 6.27e7T^{2} \)
17 \( 1 - 1.32e4iT - 4.10e8T^{2} \)
19 \( 1 - 1.11e4iT - 8.93e8T^{2} \)
23 \( 1 - 3.40e4T + 3.40e9T^{2} \)
29 \( 1 - 8.15e4iT - 1.72e10T^{2} \)
31 \( 1 - 3.89e4iT - 2.75e10T^{2} \)
37 \( 1 + 3.15e5T + 9.49e10T^{2} \)
41 \( 1 + 1.73e5iT - 1.94e11T^{2} \)
43 \( 1 - 1.94e5iT - 2.71e11T^{2} \)
47 \( 1 + 3.95e5T + 5.06e11T^{2} \)
53 \( 1 + 8.18e5iT - 1.17e12T^{2} \)
59 \( 1 + 2.08e6T + 2.48e12T^{2} \)
61 \( 1 - 1.54e6T + 3.14e12T^{2} \)
67 \( 1 + 1.57e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.66e6T + 9.09e12T^{2} \)
73 \( 1 - 8.24e5T + 1.10e13T^{2} \)
79 \( 1 - 5.41e6iT - 1.92e13T^{2} \)
83 \( 1 + 1.12e6T + 2.71e13T^{2} \)
89 \( 1 - 4.33e5iT - 4.42e13T^{2} \)
97 \( 1 + 1.13e6T + 8.07e13T^{2} \)
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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

−11.28453223008236727683858759433, −10.57328883111759405421019664730, −8.997929000649374737526284289915, −8.179073709335103970803309362527, −6.91198564628903105040022582624, −6.27323676537758320279743670696, −5.21268476404353564602597223720, −3.86740773274741297045686825876, −3.05050219547294214279370335319, −1.60600012972919082921896666471, 0.64482955257798002993496272554, 1.53324952077196346449785185973, 3.05402074470003156391686932311, 4.13472555404670895815830878355, 5.04317048477231882940864002134, 6.17231502652221587966912167583, 7.15375649613150381459881048163, 8.647646525040987158254364489030, 9.525128274555520383565704319051, 10.66907213879089346712066341952

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