Properties

Label 2-252-12.11-c3-0-27
Degree $2$
Conductor $252$
Sign $0.313 + 0.949i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.27 − 1.68i)2-s + (2.33 − 7.65i)4-s + 4.43i·5-s + 7i·7-s + (−7.56 − 21.3i)8-s + (7.46 + 10.0i)10-s + 56.3·11-s + 57.7·13-s + (11.7 + 15.9i)14-s + (−53.0 − 35.7i)16-s − 109. i·17-s − 124. i·19-s + (33.9 + 10.3i)20-s + (128. − 94.8i)22-s − 156.·23-s + ⋯
L(s)  = 1  + (0.803 − 0.594i)2-s + (0.291 − 0.956i)4-s + 0.396i·5-s + 0.377i·7-s + (−0.334 − 0.942i)8-s + (0.235 + 0.318i)10-s + 1.54·11-s + 1.23·13-s + (0.224 + 0.303i)14-s + (−0.829 − 0.558i)16-s − 1.56i·17-s − 1.49i·19-s + (0.379 + 0.115i)20-s + (1.24 − 0.918i)22-s − 1.41·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.313 + 0.949i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.313 + 0.949i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.080509299\)
\(L(\frac12)\) \(\approx\) \(3.080509299\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.27 + 1.68i)T \)
3 \( 1 \)
7 \( 1 - 7iT \)
good5 \( 1 - 4.43iT - 125T^{2} \)
11 \( 1 - 56.3T + 1.33e3T^{2} \)
13 \( 1 - 57.7T + 2.19e3T^{2} \)
17 \( 1 + 109. iT - 4.91e3T^{2} \)
19 \( 1 + 124. iT - 6.85e3T^{2} \)
23 \( 1 + 156.T + 1.21e4T^{2} \)
29 \( 1 - 210. iT - 2.43e4T^{2} \)
31 \( 1 - 120. iT - 2.97e4T^{2} \)
37 \( 1 - 299.T + 5.06e4T^{2} \)
41 \( 1 + 358. iT - 6.89e4T^{2} \)
43 \( 1 - 246. iT - 7.95e4T^{2} \)
47 \( 1 + 196.T + 1.03e5T^{2} \)
53 \( 1 - 58.3iT - 1.48e5T^{2} \)
59 \( 1 + 225.T + 2.05e5T^{2} \)
61 \( 1 - 481.T + 2.26e5T^{2} \)
67 \( 1 - 486. iT - 3.00e5T^{2} \)
71 \( 1 + 709.T + 3.57e5T^{2} \)
73 \( 1 - 866.T + 3.89e5T^{2} \)
79 \( 1 + 94.6iT - 4.93e5T^{2} \)
83 \( 1 - 208.T + 5.71e5T^{2} \)
89 \( 1 - 1.13e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.25e3T + 9.12e5T^{2} \)
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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

−11.46405277454397005145686896667, −10.84702942980209616320625525863, −9.517760172385551007031630513575, −8.849922847043546144430507949802, −6.98144406116012012769450873138, −6.31917483654728485039538613002, −5.03713170202579518495109849214, −3.83912682228541329896255581816, −2.71257304980096145088170729943, −1.11391169127287914140971669826, 1.58529953252552486511890049084, 3.77026566572415111628057928181, 4.18625371595229882269218432182, 6.03570085548785738988011719763, 6.30698405251236628775962934270, 7.928449353493023052104059204865, 8.502737540518621557397505534609, 9.822842769821283032331025296793, 11.13767636357025318254150990109, 11.98662726304470068107578980889

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