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Properties

Label 2-252-252.103-c1-0-23
Degree $2$
Conductor $252$
Sign $-0.971 + 0.235i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−0.866 + 1.5i)3-s + 2i·4-s + (−3 + 1.73i)5-s + (2.36 − 0.633i)6-s + (2.59 − 0.5i)7-s + (2 − 2i)8-s + (−1.5 − 2.59i)9-s + (4.73 + 1.26i)10-s + (−1.73 − i)11-s + (−3 − 1.73i)12-s + (−4.5 − 2.59i)13-s + (−3.09 − 2.09i)14-s − 6i·15-s − 4·16-s + (−4.5 + 2.59i)17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.499 + 0.866i)3-s + i·4-s + (−1.34 + 0.774i)5-s + (0.965 − 0.258i)6-s + (0.981 − 0.188i)7-s + (0.707 − 0.707i)8-s + (−0.5 − 0.866i)9-s + (1.49 + 0.400i)10-s + (−0.522 − 0.301i)11-s + (−0.866 − 0.499i)12-s + (−1.24 − 0.720i)13-s + (−0.827 − 0.560i)14-s − 1.54i·15-s − 16-s + (−1.09 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.971 + 0.235i$
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: \(1\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.971 + 0.235i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (0.866 - 1.5i)T \)
7 \( 1 + (-2.59 + 0.5i)T \)
good5 \( 1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.73 + i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.5 + 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.5 - 2.59i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.866 + 1.5i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.46 + 2i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.19T + 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 + 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.52 - 5.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.73T + 47T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.73T + 59T^{2} \)
61 \( 1 - 5.19iT - 61T^{2} \)
67 \( 1 - 9iT - 67T^{2} \)
71 \( 1 + 2iT - 71T^{2} \)
73 \( 1 + (10.5 - 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 3iT - 79T^{2} \)
83 \( 1 + (-2.59 - 4.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.5 - 4.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.5 - 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.31351647326737912339660476552, −10.81735978358261038575159264731, −10.04938985228442824038640900903, −8.695252479940453419913592429805, −7.86387797654245248854361540894, −6.95287434774973908929836873107, −4.96636417586955352086330618606, −4.00474979300844267647547283737, −2.81488133743212904072203226784, 0, 1.84937841768072233237056378663, 4.74205698022061536421782953276, 5.21971357394230672962812450717, 7.03293301250601471266251954068, 7.48152746099534992270864424385, 8.356146977247064240702626260333, 9.201777011342048515482415272553, 10.81527602676702048522004998949, 11.55267936016284721554525320053

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