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Properties

Label 2-252-28.19-c1-0-9
Degree $2$
Conductor $252$
Sign $0.985 - 0.168i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.553 + 1.30i)2-s + (−1.38 − 1.44i)4-s + (−0.834 − 0.481i)5-s + (1.20 − 2.35i)7-s + (2.64 − 1.00i)8-s + (1.08 − 0.819i)10-s + (4.74 − 2.74i)11-s + 3.75i·13-s + (2.40 + 2.86i)14-s + (−0.147 + 3.99i)16-s + (0.594 − 0.343i)17-s + (2.44 − 4.22i)19-s + (0.464 + 1.87i)20-s + (0.941 + 7.69i)22-s + (−1.07 − 0.620i)23-s + ⋯
L(s)  = 1  + (−0.391 + 0.920i)2-s + (−0.693 − 0.720i)4-s + (−0.373 − 0.215i)5-s + (0.453 − 0.891i)7-s + (0.934 − 0.356i)8-s + (0.344 − 0.259i)10-s + (1.43 − 0.826i)11-s + 1.04i·13-s + (0.642 + 0.766i)14-s + (−0.0369 + 0.999i)16-s + (0.144 − 0.0832i)17-s + (0.560 − 0.969i)19-s + (0.103 + 0.418i)20-s + (0.200 + 1.64i)22-s + (−0.224 − 0.129i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.985 - 0.168i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.985 - 0.168i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.996336 + 0.0843809i\)
\(L(\frac12)\) \(\approx\) \(0.996336 + 0.0843809i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.553 - 1.30i)T \)
3 \( 1 \)
7 \( 1 + (-1.20 + 2.35i)T \)
good5 \( 1 + (0.834 + 0.481i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.74 + 2.74i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.75iT - 13T^{2} \)
17 \( 1 + (-0.594 + 0.343i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.44 + 4.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.07 + 0.620i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.48T + 29T^{2} \)
31 \( 1 + (-2.41 - 4.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.36 + 2.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.42iT - 41T^{2} \)
43 \( 1 + 5.97iT - 43T^{2} \)
47 \( 1 + (1.80 - 3.13i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.04 + 3.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.34 + 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.01 - 5.20i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.17 - 4.71i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (5.76 - 3.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.22 + 0.707i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.543T + 83T^{2} \)
89 \( 1 + (0.480 + 0.277i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.8iT - 97T^{2} \)
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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

−11.84581068987178646267097385671, −11.13830389358765408292903068391, −9.909485323200777320498712782749, −8.973306357683444797682668849771, −8.170966160441832203619973504058, −7.04555652169495589658915217836, −6.33480003246770002671004200477, −4.78750703030074987767775072246, −3.92042712384677527435717677438, −1.09852598061270057493133808885, 1.65215357858530513613584263345, 3.20720915305813803930060142627, 4.39302068620939961248661987594, 5.78109117396109425914715740825, 7.41708254015858569959202623561, 8.267251692053098414794096261906, 9.303555073539004275001991471134, 10.06857409430983017317185351511, 11.22631658189282782368398364517, 12.01920274608253709725458451091

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