Properties

Label 2-252-12.11-c3-0-10
Degree $2$
Conductor $252$
Sign $-0.209 - 0.977i$
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  + (−2.36 + 1.55i)2-s + (3.14 − 7.35i)4-s + 8.56i·5-s − 7i·7-s + (4.02 + 22.2i)8-s + (−13.3 − 20.2i)10-s + 21.7·11-s − 3.31·13-s + (10.9 + 16.5i)14-s + (−44.1 − 46.2i)16-s + 12.8i·17-s + 7.76i·19-s + (62.9 + 26.9i)20-s + (−51.2 + 33.8i)22-s + 74.9·23-s + ⋯
L(s)  = 1  + (−0.834 + 0.550i)2-s + (0.393 − 0.919i)4-s + 0.765i·5-s − 0.377i·7-s + (0.177 + 0.984i)8-s + (−0.421 − 0.639i)10-s + 0.594·11-s − 0.0707·13-s + (0.208 + 0.315i)14-s + (−0.690 − 0.723i)16-s + 0.182i·17-s + 0.0937i·19-s + (0.704 + 0.301i)20-s + (−0.496 + 0.327i)22-s + 0.679·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.209 - 0.977i$
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: \((2,\ 252,\ (\ :3/2),\ -0.209 - 0.977i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.068059401\)
\(L(\frac12)\) \(\approx\) \(1.068059401\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.36 - 1.55i)T \)
3 \( 1 \)
7 \( 1 + 7iT \)
good5 \( 1 - 8.56iT - 125T^{2} \)
11 \( 1 - 21.7T + 1.33e3T^{2} \)
13 \( 1 + 3.31T + 2.19e3T^{2} \)
17 \( 1 - 12.8iT - 4.91e3T^{2} \)
19 \( 1 - 7.76iT - 6.85e3T^{2} \)
23 \( 1 - 74.9T + 1.21e4T^{2} \)
29 \( 1 - 157. iT - 2.43e4T^{2} \)
31 \( 1 - 15.8iT - 2.97e4T^{2} \)
37 \( 1 + 14.2T + 5.06e4T^{2} \)
41 \( 1 - 474. iT - 6.89e4T^{2} \)
43 \( 1 - 375. iT - 7.95e4T^{2} \)
47 \( 1 + 256.T + 1.03e5T^{2} \)
53 \( 1 - 474. iT - 1.48e5T^{2} \)
59 \( 1 + 686.T + 2.05e5T^{2} \)
61 \( 1 - 117.T + 2.26e5T^{2} \)
67 \( 1 - 421. iT - 3.00e5T^{2} \)
71 \( 1 - 5.74T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3T + 3.89e5T^{2} \)
79 \( 1 - 683. iT - 4.93e5T^{2} \)
83 \( 1 - 130.T + 5.71e5T^{2} \)
89 \( 1 + 205. iT - 7.04e5T^{2} \)
97 \( 1 - 1.71e3T + 9.12e5T^{2} \)
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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

−11.43590602076923849603856836315, −10.80783022822196133349581273179, −9.890890273382249471830917491778, −8.979625870968665297739763904847, −7.87377617783078856057271189795, −6.92584676108646121456253208690, −6.22385909638295488444814580963, −4.77245538715469078141202009115, −3.02990329015015298277657447042, −1.29643304787506084404341333042, 0.62549848065296615256650865929, 2.07332411686295679884465620794, 3.60219592277589935465680624114, 4.96457033874292038308509776723, 6.48000314021177145852650103184, 7.63144133835515191521189997873, 8.742839174479684269525300311431, 9.223211766348194581602981499455, 10.29750405934409018367809316783, 11.36067830465770058751258719552

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