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Properties

Label 2-252-252.103-c1-0-12
Degree $2$
Conductor $252$
Sign $0.998 + 0.0536i$
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  + (−1.05 + 0.943i)2-s + (−1.67 + 0.427i)3-s + (0.217 − 1.98i)4-s + (−2.49 + 1.44i)5-s + (1.36 − 2.03i)6-s + (−0.0846 − 2.64i)7-s + (1.64 + 2.29i)8-s + (2.63 − 1.43i)9-s + (1.26 − 3.87i)10-s + (−1.99 − 1.15i)11-s + (0.485 + 3.42i)12-s + (4.22 + 2.43i)13-s + (2.58 + 2.70i)14-s + (3.57 − 3.48i)15-s + (−3.90 − 0.866i)16-s + (3.54 − 2.04i)17-s + ⋯
L(s)  = 1  + (−0.744 + 0.667i)2-s + (−0.969 + 0.247i)3-s + (0.108 − 0.994i)4-s + (−1.11 + 0.644i)5-s + (0.556 − 0.830i)6-s + (−0.0320 − 0.999i)7-s + (0.582 + 0.812i)8-s + (0.877 − 0.478i)9-s + (0.401 − 1.22i)10-s + (−0.600 − 0.346i)11-s + (0.140 + 0.990i)12-s + (1.17 + 0.676i)13-s + (0.690 + 0.722i)14-s + (0.923 − 0.900i)15-s + (−0.976 − 0.216i)16-s + (0.859 − 0.496i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.998 + 0.0536i$
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: \((2,\ 252,\ (\ :1/2),\ 0.998 + 0.0536i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.485952 - 0.0130504i\)
\(L(\frac12)\) \(\approx\) \(0.485952 - 0.0130504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.05 - 0.943i)T \)
3 \( 1 + (1.67 - 0.427i)T \)
7 \( 1 + (0.0846 + 2.64i)T \)
good5 \( 1 + (2.49 - 1.44i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.99 + 1.15i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.22 - 2.43i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.54 + 2.04i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.308 + 0.534i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.29 + 4.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.811 + 1.40i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.64T + 31T^{2} \)
37 \( 1 + (-4.02 + 6.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.216 + 0.124i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.51 + 3.76i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.95T + 47T^{2} \)
53 \( 1 + (4.57 + 7.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.52T + 59T^{2} \)
61 \( 1 + 9.68iT - 61T^{2} \)
67 \( 1 - 8.17iT - 67T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + (4.66 - 2.69i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.37iT - 79T^{2} \)
83 \( 1 + (-3.53 - 6.11i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-12.0 - 6.95i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.3 + 5.99i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.42018198133887187346078053448, −11.03767772571190690076347794376, −10.36174340290359649672808466493, −9.146587239734792249556433335564, −7.81038940408874377305545713406, −7.13215643349930398883223756730, −6.26733101426991556018769501999, −4.92758762283224322820557483642, −3.69002001442977582994142029334, −0.69285368252346054045619502424, 1.19876321885668027834003053888, 3.27204391918096043530499430211, 4.72026355693440759587653820335, 5.92580641251810742123949241642, 7.48467465488002055435337009940, 8.144809834440991579819921394303, 9.145370532152531107430588262050, 10.37844660323011589379449928345, 11.26206034587708194675187936602, 11.86320909716991606197293475091

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