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Properties

Label 2-252-84.11-c1-0-5
Degree $2$
Conductor $252$
Sign $0.794 + 0.607i$
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  + (−0.627 − 1.26i)2-s + (−1.21 + 1.59i)4-s + (−2.15 + 1.24i)5-s + (2.64 + 0.0803i)7-s + (2.77 + 0.537i)8-s + (2.92 + 1.95i)10-s + (2.30 − 3.99i)11-s + 5.22·13-s + (−1.55 − 3.40i)14-s + (−1.06 − 3.85i)16-s + (4.85 + 2.80i)17-s + (−2.76 + 1.59i)19-s + (0.632 − 4.93i)20-s + (−6.50 − 0.414i)22-s + (0.359 + 0.622i)23-s + ⋯
L(s)  = 1  + (−0.443 − 0.896i)2-s + (−0.605 + 0.795i)4-s + (−0.963 + 0.556i)5-s + (0.999 + 0.0303i)7-s + (0.981 + 0.189i)8-s + (0.926 + 0.616i)10-s + (0.694 − 1.20i)11-s + 1.44·13-s + (−0.416 − 0.909i)14-s + (−0.265 − 0.964i)16-s + (1.17 + 0.680i)17-s + (−0.634 + 0.366i)19-s + (0.141 − 1.10i)20-s + (−1.38 − 0.0884i)22-s + (0.0749 + 0.129i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.924404 - 0.312952i\)
\(L(\frac12)\) \(\approx\) \(0.924404 - 0.312952i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.627 + 1.26i)T \)
3 \( 1 \)
7 \( 1 + (-2.64 - 0.0803i)T \)
good5 \( 1 + (2.15 - 1.24i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.30 + 3.99i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.22T + 13T^{2} \)
17 \( 1 + (-4.85 - 2.80i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.76 - 1.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.359 - 0.622i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.53iT - 29T^{2} \)
31 \( 1 + (-1.01 - 0.588i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.35 + 2.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.83iT - 41T^{2} \)
43 \( 1 + 11.1iT - 43T^{2} \)
47 \( 1 + (-2.70 - 4.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.79 + 1.03i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.05 - 3.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.505 - 0.874i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.9 + 6.32i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.31T + 71T^{2} \)
73 \( 1 + (4.81 - 8.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.65 - 4.41i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + (-7.38 + 4.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−11.65958545379126840608556790539, −11.01956678547530956360369942384, −10.48143234614359246119364654679, −8.778969159447296917050539057246, −8.362207449717756261353112752149, −7.35206056640432557467647005676, −5.76596565729353103384047408127, −4.01359345017091917501052249275, −3.38861694933662599109919890934, −1.35026806530563768339945420621, 1.27245732916936603556161262726, 4.11832468676270440946034829190, 4.84214174587577937106823444987, 6.22709123032550284094644069515, 7.45620259342788297767510898261, 8.137077208450301050172142381077, 8.937737837263563752362102228270, 10.03338702164257018609197609043, 11.24100414250239434503968281062, 12.00095816486724235350469369485

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