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Properties

Label 2-252-7.3-c2-0-4
Degree $2$
Conductor $252$
Sign $0.947 + 0.319i$
Analytic cond. $6.86650$
Root an. cond. $2.62040$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.29 − 4.79i)5-s + (0.5 + 6.98i)7-s + (2.29 − 3.97i)11-s − 7.84i·13-s + (16.5 + 9.58i)17-s + (−14.2 + 8.20i)19-s + (−12 − 20.7i)23-s + (33.3 − 57.8i)25-s + 10.5·29-s + (48.2 + 27.8i)31-s + (37.5 + 55.5i)35-s + (12.2 + 21.1i)37-s − 48.7i·41-s − 18.7·43-s + (10.4 − 6.00i)47-s + ⋯
L(s)  = 1  + (1.65 − 0.958i)5-s + (0.0714 + 0.997i)7-s + (0.208 − 0.361i)11-s − 0.603i·13-s + (0.976 + 0.563i)17-s + (−0.747 + 0.431i)19-s + (−0.521 − 0.903i)23-s + (1.33 − 2.31i)25-s + 0.365·29-s + (1.55 + 0.899i)31-s + (1.07 + 1.58i)35-s + (0.329 + 0.571i)37-s − 1.18i·41-s − 0.436·43-s + (0.221 − 0.127i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.947 + 0.319i$
Analytic conductor: \(6.86650\)
Root analytic conductor: \(2.62040\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1),\ 0.947 + 0.319i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.07705 - 0.340907i\)
\(L(\frac12)\) \(\approx\) \(2.07705 - 0.340907i\)
\(L(2)\) 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 \( 1 \)
7 \( 1 + (-0.5 - 6.98i)T \)
good5 \( 1 + (-8.29 + 4.79i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.29 + 3.97i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 7.84iT - 169T^{2} \)
17 \( 1 + (-16.5 - 9.58i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (14.2 - 8.20i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (12 + 20.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 10.5T + 841T^{2} \)
31 \( 1 + (-48.2 - 27.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-12.2 - 21.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 48.7iT - 1.68e3T^{2} \)
43 \( 1 + 18.7T + 1.84e3T^{2} \)
47 \( 1 + (-10.4 + 6.00i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (18.7 - 32.3i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (54.7 + 31.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (96.3 - 55.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (31.9 - 55.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 21.5T + 5.04e3T^{2} \)
73 \( 1 + (53.5 + 30.9i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-1.90 - 3.30i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 100. iT - 6.88e3T^{2} \)
89 \( 1 + (85.7 - 49.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 63.5iT - 9.40e3T^{2} \)
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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

−12.24077420808101107865011671036, −10.49638907712498876768114615572, −9.887726122683087931206827117344, −8.780659052264115758423769790370, −8.289534559162650752027813658698, −6.25313098623726921985992720796, −5.78011661490837274312164950862, −4.71872780612661998229777532749, −2.70870833627045664897892853853, −1.40172365567159374842191594933, 1.61406177302383219656971642314, 2.98542782511905812272260297336, 4.57835897958164650412157643913, 5.99436355303547873342819732713, 6.73285556157010210341119547796, 7.72250732824751483141682994967, 9.403717536230121545184670235075, 9.923492804322559799728657039164, 10.71619174864060656341652841991, 11.70892584585634166716459092278

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