Properties

Label 2-252-252.103-c1-0-21
Degree $2$
Conductor $252$
Sign $0.726 - 0.687i$
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  + (0.248 + 1.39i)2-s + (1.22 − 1.22i)3-s + (−1.87 + 0.691i)4-s + (0.705 − 0.407i)5-s + (2.01 + 1.39i)6-s + (2.03 + 1.69i)7-s + (−1.42 − 2.44i)8-s + (−0.0104 − 2.99i)9-s + (0.742 + 0.881i)10-s + (4.17 + 2.41i)11-s + (−1.44 + 3.14i)12-s + (−2.20 − 1.27i)13-s + (−1.85 + 3.25i)14-s + (0.363 − 1.36i)15-s + (3.04 − 2.59i)16-s + (−0.503 + 0.290i)17-s + ⋯
L(s)  = 1  + (0.175 + 0.984i)2-s + (0.705 − 0.708i)3-s + (−0.938 + 0.345i)4-s + (0.315 − 0.182i)5-s + (0.821 + 0.570i)6-s + (0.768 + 0.639i)7-s + (−0.505 − 0.862i)8-s + (−0.00349 − 0.999i)9-s + (0.234 + 0.278i)10-s + (1.25 + 0.727i)11-s + (−0.417 + 0.908i)12-s + (−0.611 − 0.352i)13-s + (−0.494 + 0.869i)14-s + (0.0937 − 0.352i)15-s + (0.760 − 0.649i)16-s + (−0.122 + 0.0704i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.726 - 0.687i$
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: \((2,\ 252,\ (\ :1/2),\ 0.726 - 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58311 + 0.630324i\)
\(L(\frac12)\) \(\approx\) \(1.58311 + 0.630324i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.248 - 1.39i)T \)
3 \( 1 + (-1.22 + 1.22i)T \)
7 \( 1 + (-2.03 - 1.69i)T \)
good5 \( 1 + (-0.705 + 0.407i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.17 - 2.41i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.20 + 1.27i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.503 - 0.290i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.24 - 3.88i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.60 + 3.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.16 + 5.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.62T + 31T^{2} \)
37 \( 1 + (-1.62 + 2.82i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.04 + 1.75i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.72 - 2.72i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.58T + 47T^{2} \)
53 \( 1 + (1.09 + 1.89i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 7.24iT - 61T^{2} \)
67 \( 1 + 5.02iT - 67T^{2} \)
71 \( 1 + 3.64iT - 71T^{2} \)
73 \( 1 + (-2.26 + 1.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 10.1iT - 79T^{2} \)
83 \( 1 + (0.820 + 1.42i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.90 + 1.10i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.14 - 3.54i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.55550019222914989705326141742, −11.55652639243979829234105988894, −9.676859340659734922936444988983, −9.042507862770446773223959035828, −8.136371002473016491020710506874, −7.24675431136078254042011256357, −6.27635785758002985100765476035, −5.11238739870475994622790330596, −3.76199037088527553402592614865, −1.86986796063668871620819399512, 1.77191666219599417344650095568, 3.28746783917238440500783630903, 4.29324494649521324200225600337, 5.27730633654432219037682855687, 7.05344355443065749402496550193, 8.527196299614265081110245830285, 9.170793865579632375710453112713, 10.07061638208549708638287526459, 11.09593739712045691011282871345, 11.50176484928760753144894192643

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