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Properties

Label 2-252-252.31-c1-0-3
Degree $2$
Conductor $252$
Sign $-0.805 - 0.592i$
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  + (0.447 + 1.34i)2-s + (−0.385 − 1.68i)3-s + (−1.59 + 1.20i)4-s + 3.94i·5-s + (2.09 − 1.27i)6-s + (−2.37 − 1.16i)7-s + (−2.32 − 1.60i)8-s + (−2.70 + 1.30i)9-s + (−5.29 + 1.76i)10-s + 4.89i·11-s + (2.64 + 2.23i)12-s + (1.13 + 0.657i)13-s + (0.506 − 3.70i)14-s + (6.66 − 1.52i)15-s + (1.11 − 3.84i)16-s + (1.67 + 0.969i)17-s + ⋯
L(s)  = 1  + (0.316 + 0.948i)2-s + (−0.222 − 0.974i)3-s + (−0.799 + 0.600i)4-s + 1.76i·5-s + (0.854 − 0.519i)6-s + (−0.897 − 0.441i)7-s + (−0.822 − 0.568i)8-s + (−0.900 + 0.434i)9-s + (−1.67 + 0.558i)10-s + 1.47i·11-s + (0.763 + 0.645i)12-s + (0.315 + 0.182i)13-s + (0.135 − 0.990i)14-s + (1.72 − 0.393i)15-s + (0.278 − 0.960i)16-s + (0.407 + 0.235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.805 - 0.592i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.288301 + 0.878814i\)
\(L(\frac12)\) \(\approx\) \(0.288301 + 0.878814i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.447 - 1.34i)T \)
3 \( 1 + (0.385 + 1.68i)T \)
7 \( 1 + (2.37 + 1.16i)T \)
good5 \( 1 - 3.94iT - 5T^{2} \)
11 \( 1 - 4.89iT - 11T^{2} \)
13 \( 1 + (-1.13 - 0.657i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.67 - 0.969i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.49 - 2.58i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.37iT - 23T^{2} \)
29 \( 1 + (1.32 + 2.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.443 + 0.767i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.237 - 0.410i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.79 - 5.07i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.31 + 1.33i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.23 - 2.13i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.34 + 10.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.49 - 6.05i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.05 - 1.18i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.20 - 0.697i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.15iT - 71T^{2} \)
73 \( 1 + (-3.96 - 2.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.70 + 5.02i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.29 - 2.24i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (11.6 - 6.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.88 + 1.08i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.65339256664092195034539588255, −11.71284006993028952994549674534, −10.43938371469956095025839452166, −9.598360944066018056125210380700, −7.926882917193390303378637394228, −7.16533022469820733451606152981, −6.65188341544607301511380588437, −5.82059910242328164085605011865, −3.94973446282685644066522398967, −2.66155358130647912450826668641, 0.69113251824692015414990187020, 3.11425283148658291405310081227, 4.13328972195577406404073776059, 5.42188469210297703930632535481, 5.76951877326115371546012531291, 8.429498045031781981319267203139, 9.098082667196725777551606979012, 9.601681248715601618239513705506, 10.83280700526758369358165609592, 11.69982325886131243822403730924

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