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Properties

Label 2-252-252.23-c1-0-34
Degree $2$
Conductor $252$
Sign $0.495 + 0.868i$
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  + (−1.35 + 0.387i)2-s + (0.672 − 1.59i)3-s + (1.69 − 1.05i)4-s + (3.19 − 1.84i)5-s + (−0.294 + 2.43i)6-s + (2.29 − 1.31i)7-s + (−1.90 + 2.09i)8-s + (−2.09 − 2.14i)9-s + (−3.62 + 3.74i)10-s + (−1.79 + 3.11i)11-s + (−0.542 − 3.42i)12-s + (−0.565 + 0.978i)13-s + (−2.60 + 2.68i)14-s + (−0.795 − 6.33i)15-s + (1.77 − 3.58i)16-s + (−5.61 + 3.23i)17-s + ⋯
L(s)  = 1  + (−0.961 + 0.274i)2-s + (0.388 − 0.921i)3-s + (0.849 − 0.527i)4-s + (1.42 − 0.824i)5-s + (−0.120 + 0.992i)6-s + (0.866 − 0.498i)7-s + (−0.672 + 0.740i)8-s + (−0.698 − 0.715i)9-s + (−1.14 + 1.18i)10-s + (−0.542 + 0.939i)11-s + (−0.156 − 0.987i)12-s + (−0.156 + 0.271i)13-s + (−0.696 + 0.717i)14-s + (−0.205 − 1.63i)15-s + (0.443 − 0.896i)16-s + (−1.36 + 0.785i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.495 + 0.868i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.495 + 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01654 - 0.590458i\)
\(L(\frac12)\) \(\approx\) \(1.01654 - 0.590458i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 0.387i)T \)
3 \( 1 + (-0.672 + 1.59i)T \)
7 \( 1 + (-2.29 + 1.31i)T \)
good5 \( 1 + (-3.19 + 1.84i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.79 - 3.11i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.565 - 0.978i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.61 - 3.23i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.20 - 2.42i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.522 + 0.904i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.32 + 0.764i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.45iT - 31T^{2} \)
37 \( 1 + (-1.02 + 1.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.782 + 0.451i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.77 - 3.33i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.71T + 47T^{2} \)
53 \( 1 + (-0.975 + 0.563i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.835T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 - 2.47T + 71T^{2} \)
73 \( 1 + (-3.43 - 5.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 6.62iT - 79T^{2} \)
83 \( 1 + (5.37 + 9.31i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-15.6 - 9.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.46 + 9.45i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.92165089704878459519059665184, −10.67433302970902260092652747011, −9.746157503276338236311091063846, −8.885505439576615140839249114423, −8.070400820529570770580423166543, −7.08438215732420748827008595184, −6.07198660727577187710774071051, −4.93378392859704576878233526612, −2.20676139515713090731554530046, −1.45359997281519475691567030572, 2.24795558050462235016973689201, 3.02108336215436471968444569735, 5.10324818944600507080182166643, 6.15150826122337824150700411505, 7.53923835930453404975843817884, 8.714580583770084983804614128185, 9.342074325262561398868338092683, 10.24977933181276060981789221201, 10.97422604626714075621582438909, 11.59285893647258132919970810097

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