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Properties

Label 2-252-252.103-c1-0-25
Degree $2$
Conductor $252$
Sign $0.760 + 0.649i$
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  + (−1.15 + 0.809i)2-s + (0.630 − 1.61i)3-s + (0.688 − 1.87i)4-s + (2.22 − 1.28i)5-s + (0.575 + 2.38i)6-s + (2.01 + 1.71i)7-s + (0.722 + 2.73i)8-s + (−2.20 − 2.03i)9-s + (−1.53 + 3.29i)10-s + (−4.82 − 2.78i)11-s + (−2.59 − 2.29i)12-s + (3.36 + 1.94i)13-s + (−3.72 − 0.356i)14-s + (−0.670 − 4.39i)15-s + (−3.05 − 2.58i)16-s + (2.65 − 1.53i)17-s + ⋯
L(s)  = 1  + (−0.819 + 0.572i)2-s + (0.363 − 0.931i)3-s + (0.344 − 0.938i)4-s + (0.994 − 0.574i)5-s + (0.235 + 0.971i)6-s + (0.761 + 0.648i)7-s + (0.255 + 0.966i)8-s + (−0.735 − 0.677i)9-s + (−0.486 + 1.04i)10-s + (−1.45 − 0.839i)11-s + (−0.749 − 0.662i)12-s + (0.933 + 0.538i)13-s + (−0.995 − 0.0952i)14-s + (−0.173 − 1.13i)15-s + (−0.762 − 0.646i)16-s + (0.642 − 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.760 + 0.649i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.760 + 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06376 - 0.392134i\)
\(L(\frac12)\) \(\approx\) \(1.06376 - 0.392134i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 - 0.809i)T \)
3 \( 1 + (-0.630 + 1.61i)T \)
7 \( 1 + (-2.01 - 1.71i)T \)
good5 \( 1 + (-2.22 + 1.28i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.82 + 2.78i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.36 - 1.94i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.65 + 1.53i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.51 + 2.62i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.90 - 2.25i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.91 - 5.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.712T + 31T^{2} \)
37 \( 1 + (-3.16 + 5.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.66 + 3.27i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.877 + 0.506i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.374T + 47T^{2} \)
53 \( 1 + (2.00 + 3.47i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.31T + 59T^{2} \)
61 \( 1 - 9.20iT - 61T^{2} \)
67 \( 1 - 6.34iT - 67T^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 + (13.8 - 8.00i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 6.94iT - 79T^{2} \)
83 \( 1 + (-4.89 - 8.48i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.02 - 1.16i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.10 - 1.79i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.82495775243025238761385324061, −10.94023283806919522654841345357, −9.690697788472842137544696621566, −8.689962575335827992671072453790, −8.270960921059030054877401456966, −7.14755897941310549955811389191, −5.75984721850268311761407246208, −5.42324646382300231841765450234, −2.53941435454327172447861028843, −1.31186952192784816837409981740, 2.02338987685295765518909105627, 3.24928481175603533245848717903, 4.65582626532629830149035670717, 6.07765257956743401584583874503, 7.79967770176710374373373385821, 8.202593522530859915711896839653, 9.693802643707665152086756154682, 10.40607275128493167333168834157, 10.51465383318135695421794973799, 11.80514736731040258233751409935

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