Properties

Label 2-252-12.11-c7-0-29
Degree $2$
Conductor $252$
Sign $-0.276 - 0.961i$
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  + (6.55 + 9.22i)2-s + (−42.1 + 120. i)4-s − 328. i·5-s + 343i·7-s + (−1.39e3 + 403. i)8-s + (3.02e3 − 2.15e3i)10-s + 3.62e3·11-s − 7.66e3·13-s + (−3.16e3 + 2.24e3i)14-s + (−1.28e4 − 1.01e4i)16-s − 1.44e4i·17-s + 2.75e4i·19-s + (3.96e4 + 1.38e4i)20-s + (2.37e4 + 3.34e4i)22-s + 5.37e4·23-s + ⋯
L(s)  = 1  + (0.579 + 0.815i)2-s + (−0.329 + 0.944i)4-s − 1.17i·5-s + 0.377i·7-s + (−0.960 + 0.278i)8-s + (0.957 − 0.680i)10-s + 0.821·11-s − 0.967·13-s + (−0.308 + 0.218i)14-s + (−0.783 − 0.621i)16-s − 0.711i·17-s + 0.921i·19-s + (1.10 + 0.386i)20-s + (0.475 + 0.669i)22-s + 0.921·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.276 - 0.961i$
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: \((2,\ 252,\ (\ :7/2),\ -0.276 - 0.961i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.399907707\)
\(L(\frac12)\) \(\approx\) \(2.399907707\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-6.55 - 9.22i)T \)
3 \( 1 \)
7 \( 1 - 343iT \)
good5 \( 1 + 328. iT - 7.81e4T^{2} \)
11 \( 1 - 3.62e3T + 1.94e7T^{2} \)
13 \( 1 + 7.66e3T + 6.27e7T^{2} \)
17 \( 1 + 1.44e4iT - 4.10e8T^{2} \)
19 \( 1 - 2.75e4iT - 8.93e8T^{2} \)
23 \( 1 - 5.37e4T + 3.40e9T^{2} \)
29 \( 1 + 1.69e5iT - 1.72e10T^{2} \)
31 \( 1 - 1.67e5iT - 2.75e10T^{2} \)
37 \( 1 - 1.63e5T + 9.49e10T^{2} \)
41 \( 1 - 5.51e5iT - 1.94e11T^{2} \)
43 \( 1 - 2.92e5iT - 2.71e11T^{2} \)
47 \( 1 - 6.12e5T + 5.06e11T^{2} \)
53 \( 1 - 2.12e6iT - 1.17e12T^{2} \)
59 \( 1 - 1.31e6T + 2.48e12T^{2} \)
61 \( 1 - 2.01e4T + 3.14e12T^{2} \)
67 \( 1 - 3.50e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.43e6T + 9.09e12T^{2} \)
73 \( 1 - 4.70e6T + 1.10e13T^{2} \)
79 \( 1 - 2.64e6iT - 1.92e13T^{2} \)
83 \( 1 + 5.60e6T + 2.71e13T^{2} \)
89 \( 1 - 3.13e6iT - 4.42e13T^{2} \)
97 \( 1 + 9.17e6T + 8.07e13T^{2} \)
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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

−11.53666673879004954309951938859, −9.701328103514330261794164752148, −8.997152926641407310540722632348, −8.108430201879479559323202924763, −7.07823722066058029880995768425, −5.91304594861409028288711568346, −4.97633678962920892771015934775, −4.19957486621936070797699412166, −2.72429664657515316473587068600, −1.00426715112369641368549226253, 0.53358301557335327272814253937, 2.00429640461106962254933334782, 3.05137969779349233141419663793, 4.01366895782099867472963689573, 5.21558870959933277411158257693, 6.54390468091160994909669922303, 7.19408370276569074564275573682, 8.901409234542517502985704378896, 9.859707816395554038937346361490, 10.74400046480296304274954228330

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