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Properties

Label 2-252-7.4-c3-0-0
Degree $2$
Conductor $252$
Sign $-0.999 - 0.0316i$
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  + (4.91 + 8.50i)5-s + (−16.3 − 8.74i)7-s + (−7.08 + 12.2i)11-s − 26.1·13-s + (−39.2 + 68.0i)17-s + (−36.5 − 63.3i)19-s + (−48 − 83.1i)23-s + (14.2 − 24.6i)25-s − 173.·29-s + (−33.6 + 58.2i)31-s + (−5.77 − 181. i)35-s + (150. + 261. i)37-s − 472.·41-s − 463.·43-s + (−45.5 − 78.9i)47-s + ⋯
L(s)  = 1  + (0.439 + 0.761i)5-s + (−0.881 − 0.472i)7-s + (−0.194 + 0.336i)11-s − 0.557·13-s + (−0.560 + 0.971i)17-s + (−0.441 − 0.765i)19-s + (−0.435 − 0.753i)23-s + (0.113 − 0.197i)25-s − 1.10·29-s + (−0.194 + 0.337i)31-s + (−0.0278 − 0.878i)35-s + (0.670 + 1.16i)37-s − 1.79·41-s − 1.64·43-s + (−0.141 − 0.245i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0316i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.999 - 0.0316i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.999 - 0.0316i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2416686376\)
\(L(\frac12)\) \(\approx\) \(0.2416686376\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.3 + 8.74i)T \)
good5 \( 1 + (-4.91 - 8.50i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (7.08 - 12.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 26.1T + 2.19e3T^{2} \)
17 \( 1 + (39.2 - 68.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (36.5 + 63.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (48 + 83.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 173.T + 2.43e4T^{2} \)
31 \( 1 + (33.6 - 58.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-150. - 261. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 472.T + 6.89e4T^{2} \)
43 \( 1 + 463.T + 7.95e4T^{2} \)
47 \( 1 + (45.5 + 78.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (81.6 - 141. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (300. - 520. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (285. + 495. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-269. + 467. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.06e3T + 3.57e5T^{2} \)
73 \( 1 + (-221. + 383. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-22.8 - 39.5i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 686.T + 5.71e5T^{2} \)
89 \( 1 + (-330. - 571. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 658.T + 9.12e5T^{2} \)
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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

−12.15711747290948594448601763489, −10.83905735156482944929392012784, −10.25475862821861589623596595033, −9.395683338394663266537671233916, −8.126736002717176302962911015226, −6.80830707133902763410965136449, −6.37804541271960320505743821693, −4.78743006268608402040249205001, −3.41427486648488404713490255534, −2.15766497628013387519010378396, 0.087113658155435372054388344910, 2.01257568841959446359309684880, 3.48945111338199670147891375548, 5.03666387040549919646068969394, 5.88650312182277215613492760484, 7.07119791041385227407558123198, 8.339931259615206987548378838299, 9.353322481263941553952171158481, 9.857014990756946236681375827169, 11.22563346070629997813349011781

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