Properties

Label 2-252-7.2-c3-0-5
Degree $2$
Conductor $252$
Sign $0.965 + 0.261i$
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  + (2.58 − 4.47i)5-s + (18.1 − 3.60i)7-s + (29.2 + 50.7i)11-s − 38.3·13-s + (−26.3 − 45.6i)17-s + (65.1 − 112. i)19-s + (38.2 − 66.3i)23-s + (49.1 + 85.1i)25-s + 288.·29-s + (70.2 + 121. i)31-s + (30.7 − 90.5i)35-s + (41.2 − 71.4i)37-s + 282.·41-s − 172·43-s + (66.6 − 115. i)47-s + ⋯
L(s)  = 1  + (0.231 − 0.400i)5-s + (0.980 − 0.194i)7-s + (0.802 + 1.39i)11-s − 0.817·13-s + (−0.375 − 0.650i)17-s + (0.786 − 1.36i)19-s + (0.347 − 0.601i)23-s + (0.393 + 0.681i)25-s + 1.84·29-s + (0.407 + 0.705i)31-s + (0.148 − 0.437i)35-s + (0.183 − 0.317i)37-s + 1.07·41-s − 0.609·43-s + (0.206 − 0.358i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.965 + 0.261i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.965 + 0.261i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.155074629\)
\(L(\frac12)\) \(\approx\) \(2.155074629\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-18.1 + 3.60i)T \)
good5 \( 1 + (-2.58 + 4.47i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-29.2 - 50.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 38.3T + 2.19e3T^{2} \)
17 \( 1 + (26.3 + 45.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-65.1 + 112. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-38.2 + 66.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 288.T + 2.43e4T^{2} \)
31 \( 1 + (-70.2 - 121. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-41.2 + 71.4i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 282.T + 6.89e4T^{2} \)
43 \( 1 + 172T + 7.95e4T^{2} \)
47 \( 1 + (-66.6 + 115. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-18.2 - 31.5i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (126. + 219. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (249. - 432. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (105. + 182. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 107.T + 3.57e5T^{2} \)
73 \( 1 + (180. + 312. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-417. + 723. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 731.T + 5.71e5T^{2} \)
89 \( 1 + (108. - 188. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 252.T + 9.12e5T^{2} \)
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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

−11.72048741689611219795588712789, −10.60703146158966047398558162850, −9.525473644346209384916256801288, −8.810543672004031536094753328769, −7.46161381670112291694113453459, −6.77687890553550452350249835517, −4.93999339069085495119706031777, −4.61302477518302056427244929586, −2.55981552287030904010371158090, −1.11828231992224251567048809897, 1.24593130288034270956631095977, 2.83898607292261422340164062979, 4.25880925932733670737552323964, 5.59114278170465606965992374181, 6.50409511627148242776915868371, 7.86706128157970638772533027491, 8.595337859441716474796987349437, 9.788439795624705366908872778677, 10.77369639263995943357935030916, 11.61982879943308561349733278082

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