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Properties

Label 2-252-63.25-c1-0-5
Degree $2$
Conductor $252$
Sign $0.992 + 0.123i$
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  + (1.40 + 1.01i)3-s + (1.26 − 2.18i)5-s + (0.527 − 2.59i)7-s + (0.929 + 2.85i)9-s + (0.687 + 1.18i)11-s + (−2.80 − 4.84i)13-s + (3.98 − 1.77i)15-s + (−2.69 + 4.66i)17-s + (2.44 + 4.23i)19-s + (3.37 − 3.09i)21-s + (−2.08 + 3.61i)23-s + (−0.675 − 1.17i)25-s + (−1.59 + 4.94i)27-s + (−1.56 + 2.71i)29-s + 4.80·31-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + (0.563 − 0.976i)5-s + (0.199 − 0.979i)7-s + (0.309 + 0.950i)9-s + (0.207 + 0.358i)11-s + (−0.776 − 1.34i)13-s + (1.02 − 0.458i)15-s + (−0.653 + 1.13i)17-s + (0.561 + 0.972i)19-s + (0.737 − 0.675i)21-s + (−0.435 + 0.753i)23-s + (−0.135 − 0.234i)25-s + (−0.307 + 0.951i)27-s + (−0.291 + 0.504i)29-s + 0.862·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.992 + 0.123i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.992 + 0.123i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70651 - 0.105713i\)
\(L(\frac12)\) \(\approx\) \(1.70651 - 0.105713i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 - 1.01i)T \)
7 \( 1 + (-0.527 + 2.59i)T \)
good5 \( 1 + (-1.26 + 2.18i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.687 - 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.80 + 4.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.69 - 4.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.44 - 4.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.08 - 3.61i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.56 - 2.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.80T + 31T^{2} \)
37 \( 1 + (2.69 + 4.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.02 + 5.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.44 + 4.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + (7.00 - 12.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 + 6.85T + 61T^{2} \)
67 \( 1 + 8.11T + 67T^{2} \)
71 \( 1 + 2.25T + 71T^{2} \)
73 \( 1 + (-3.51 + 6.08i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.75T + 79T^{2} \)
83 \( 1 + (-7.48 + 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.75 - 4.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.894 + 1.54i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.27688277358169061763125593019, −10.64481684302294243281524778658, −10.10097058016603644060379631762, −9.201310470984201228743999139842, −8.176731130295130772758346549755, −7.40910001314107042243908228944, −5.63109427696903961476214451144, −4.63438460376984928641611701215, −3.52973001635413369069259241409, −1.70353336615132151342990655443, 2.18069343014687997377646021356, 2.90992512117847082291694465823, 4.75741393601870881949639477081, 6.43948428844415235659403337546, 6.87700902365095377839148909776, 8.215454008709784707361882995436, 9.238405438628068940864276946963, 9.812505533761908266963343086517, 11.43170052373315021276806897638, 11.88689415700405606476762680453

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