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Properties

Label 2-252-9.4-c1-0-5
Degree $2$
Conductor $252$
Sign $0.549 + 0.835i$
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  + (1.29 − 1.15i)3-s + (0.555 − 0.962i)5-s + (−0.5 − 0.866i)7-s + (0.349 − 2.97i)9-s + (0.944 + 1.63i)11-s + (0.5 − 0.866i)13-s + (−0.388 − 1.88i)15-s − 5.87·17-s + 7.09·19-s + (−1.64 − 0.545i)21-s + (−1.99 + 3.45i)23-s + (1.88 + 3.26i)25-s + (−2.97 − 4.25i)27-s + (−0.493 − 0.855i)29-s + (0.333 − 0.576i)31-s + ⋯
L(s)  = 1  + (0.747 − 0.664i)3-s + (0.248 − 0.430i)5-s + (−0.188 − 0.327i)7-s + (0.116 − 0.993i)9-s + (0.284 + 0.493i)11-s + (0.138 − 0.240i)13-s + (−0.100 − 0.486i)15-s − 1.42·17-s + 1.62·19-s + (−0.358 − 0.118i)21-s + (−0.415 + 0.720i)23-s + (0.376 + 0.652i)25-s + (−0.572 − 0.819i)27-s + (−0.0916 − 0.158i)29-s + (0.0598 − 0.103i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.549 + 0.835i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.549 + 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40009 - 0.755387i\)
\(L(\frac12)\) \(\approx\) \(1.40009 - 0.755387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.29 + 1.15i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.555 + 0.962i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.944 - 1.63i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.87T + 17T^{2} \)
19 \( 1 - 7.09T + 19T^{2} \)
23 \( 1 + (1.99 - 3.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.493 + 0.855i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.333 + 0.576i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.33T + 37T^{2} \)
41 \( 1 + (-0.944 + 1.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.43 - 9.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.54 - 9.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + (2.38 - 4.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.88 + 3.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.04 - 3.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + 3.09T + 73T^{2} \)
79 \( 1 + (3.21 + 5.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.93 - 10.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + (0.382 + 0.662i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.08798494847799151559971714861, −11.04382100757040551677770838146, −9.566336990927589354699633633018, −9.158793996244397911710173855969, −7.86675533446428163723688672032, −7.10275848746921385518080285658, −5.94018485318890720140628828699, −4.42135656551607590398068520042, −3.03166646473901264076245437642, −1.44567093378340039827623271809, 2.34944387046981384583710587024, 3.51998903137177942048192118988, 4.79467338124890664422881234876, 6.13654817495053243056019956516, 7.30792341746388548027589971490, 8.597576973751347742171896637601, 9.209222057727912441260642836649, 10.24657414469675309870697009346, 11.05005178271490672874141437349, 12.11734272101163571222464582417

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