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Properties

Label 2-252-63.58-c1-0-2
Degree $2$
Conductor $252$
Sign $0.752 + 0.658i$
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  + (−1.20 − 1.24i)3-s + (0.951 + 1.64i)5-s + (2.11 − 1.58i)7-s + (−0.0910 + 2.99i)9-s + (1.53 − 2.65i)11-s + (1.13 − 1.96i)13-s + (0.901 − 3.17i)15-s + (−0.713 − 1.23i)17-s + (2.98 − 5.16i)19-s + (−4.52 − 0.712i)21-s + (3.57 + 6.19i)23-s + (0.689 − 1.19i)25-s + (3.83 − 3.50i)27-s + (0.468 + 0.810i)29-s − 8.22·31-s + ⋯
L(s)  = 1  + (−0.696 − 0.717i)3-s + (0.425 + 0.737i)5-s + (0.799 − 0.600i)7-s + (−0.0303 + 0.999i)9-s + (0.462 − 0.800i)11-s + (0.313 − 0.543i)13-s + (0.232 − 0.818i)15-s + (−0.173 − 0.299i)17-s + (0.684 − 1.18i)19-s + (−0.987 − 0.155i)21-s + (0.746 + 1.29i)23-s + (0.137 − 0.238i)25-s + (0.738 − 0.674i)27-s + (0.0869 + 0.150i)29-s − 1.47·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.752 + 0.658i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.752 + 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10744 - 0.416027i\)
\(L(\frac12)\) \(\approx\) \(1.10744 - 0.416027i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 + 1.24i)T \)
7 \( 1 + (-2.11 + 1.58i)T \)
good5 \( 1 + (-0.951 - 1.64i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.53 + 2.65i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.13 + 1.96i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.713 + 1.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.98 + 5.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.57 - 6.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.468 - 0.810i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.22T + 31T^{2} \)
37 \( 1 + (1.41 - 2.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.31 - 9.20i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.98 - 5.16i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.966T + 47T^{2} \)
53 \( 1 + (-5.45 - 9.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 0.899T + 61T^{2} \)
67 \( 1 - 1.62T + 67T^{2} \)
71 \( 1 + 2.36T + 71T^{2} \)
73 \( 1 + (0.996 + 1.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 8.33T + 79T^{2} \)
83 \( 1 + (7.98 + 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.58 - 4.48i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.922 - 1.59i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.55071729870519826118344047485, −11.20586253487778575287078010698, −10.37162249994863433461228732130, −8.989038719326512536809567986173, −7.69718332564724649550725252545, −6.97450232708130235307655587894, −5.92703545634463901081305607697, −4.88443858813265163558101026662, −3.06106403327273518282555859299, −1.26849478989146811271247865616, 1.68838356305210669205885573198, 3.94634766252142764977194749775, 5.01099494595401814255461089866, 5.74612323633087442264182233819, 7.04612976707767677903507782819, 8.648509857060089761399493021826, 9.211169545253614929381572113566, 10.27193810931700569840522499749, 11.22545594240884528157041755496, 12.15686215010383041722953112265

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