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Properties

Label 2-252-63.41-c1-0-4
Degree $2$
Conductor $252$
Sign $0.505 + 0.862i$
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  + (−1.72 + 0.136i)3-s + (−0.276 + 0.479i)5-s + (−0.519 − 2.59i)7-s + (2.96 − 0.472i)9-s + (4.03 − 2.32i)11-s + (−3.58 − 2.06i)13-s + (0.412 − 0.866i)15-s + 7.24·17-s − 6.71i·19-s + (1.25 + 4.40i)21-s + (−4.85 − 2.80i)23-s + (2.34 + 4.06i)25-s + (−5.05 + 1.22i)27-s + (1.16 − 0.673i)29-s + (0.830 + 0.479i)31-s + ⋯
L(s)  = 1  + (−0.996 + 0.0789i)3-s + (−0.123 + 0.214i)5-s + (−0.196 − 0.980i)7-s + (0.987 − 0.157i)9-s + (1.21 − 0.702i)11-s + (−0.993 − 0.573i)13-s + (0.106 − 0.223i)15-s + 1.75·17-s − 1.54i·19-s + (0.273 + 0.962i)21-s + (−1.01 − 0.584i)23-s + (0.469 + 0.812i)25-s + (−0.972 + 0.234i)27-s + (0.216 − 0.125i)29-s + (0.149 + 0.0861i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.505 + 0.862i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.505 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.751574 - 0.430647i\)
\(L(\frac12)\) \(\approx\) \(0.751574 - 0.430647i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 0.136i)T \)
7 \( 1 + (0.519 + 2.59i)T \)
good5 \( 1 + (0.276 - 0.479i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.03 + 2.32i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.58 + 2.06i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.24T + 17T^{2} \)
19 \( 1 + 6.71iT - 19T^{2} \)
23 \( 1 + (4.85 + 2.80i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.16 + 0.673i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.830 - 0.479i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.06T + 37T^{2} \)
41 \( 1 + (-2.39 + 4.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.02 + 1.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.90 - 8.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.43iT - 53T^{2} \)
59 \( 1 + (3.89 - 6.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.37 + 3.10i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.68 + 2.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.407iT - 71T^{2} \)
73 \( 1 + 8.63iT - 73T^{2} \)
79 \( 1 + (0.318 + 0.551i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.78 + 4.82i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.93T + 89T^{2} \)
97 \( 1 + (7.48 - 4.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.90115523736752825284307673100, −10.88306431881156429842812602249, −10.21057406005218591608560893280, −9.232569208438263391266005903862, −7.61846175850640308078564850454, −6.88874623421889013198484683675, −5.80747072646104603077950516702, −4.60420367296183870441598428928, −3.39159804466724778781459583111, −0.850831287254953838118291700032, 1.73423419363807500988602705783, 3.86570875453293317299675706019, 5.14687901970472982111926672870, 6.05145366158321345330393461047, 7.07514328309206071827881351648, 8.244679876904642614425367883362, 9.712423798556694360114407937218, 10.02626261734644334231493498607, 11.72072528403728916702474280662, 12.14585664551286656818752522433

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