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Properties

Label 2-252-63.20-c3-0-22
Degree $2$
Conductor $252$
Sign $-0.436 - 0.899i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.09 + 1.03i)3-s + (−10.5 − 18.2i)5-s + (−4.11 − 18.0i)7-s + (24.8 − 10.5i)9-s + (−21.5 − 12.4i)11-s + (−52.5 + 30.3i)13-s + (72.7 + 82.1i)15-s + 117.·17-s − 104. i·19-s + (39.6 + 87.6i)21-s + (−17.4 + 10.0i)23-s + (−160. + 277. i)25-s + (−115. + 79.5i)27-s + (−24.2 − 14.0i)29-s + (−216. + 125. i)31-s + ⋯
L(s)  = 1  + (−0.979 + 0.199i)3-s + (−0.944 − 1.63i)5-s + (−0.222 − 0.974i)7-s + (0.920 − 0.390i)9-s + (−0.589 − 0.340i)11-s + (−1.12 + 0.647i)13-s + (1.25 + 1.41i)15-s + 1.67·17-s − 1.26i·19-s + (0.412 + 0.911i)21-s + (−0.158 + 0.0915i)23-s + (−1.28 + 2.22i)25-s + (−0.823 + 0.566i)27-s + (−0.155 − 0.0897i)29-s + (−1.25 + 0.724i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.436 - 0.899i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.436 - 0.899i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1280688835\)
\(L(\frac12)\) \(\approx\) \(0.1280688835\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.09 - 1.03i)T \)
7 \( 1 + (4.11 + 18.0i)T \)
good5 \( 1 + (10.5 + 18.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (21.5 + 12.4i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (52.5 - 30.3i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 117.T + 4.91e3T^{2} \)
19 \( 1 + 104. iT - 6.85e3T^{2} \)
23 \( 1 + (17.4 - 10.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (24.2 + 14.0i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (216. - 125. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 18.2T + 5.06e4T^{2} \)
41 \( 1 + (-153. - 265. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-74.5 + 129. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-108. + 188. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 116. iT - 1.48e5T^{2} \)
59 \( 1 + (38.3 + 66.4i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-493. - 285. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (33.8 + 58.6i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 796. iT - 3.57e5T^{2} \)
73 \( 1 + 710. iT - 3.89e5T^{2} \)
79 \( 1 + (40.0 - 69.3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (57.6 - 99.8i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + (-444. - 256. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.13611803554707606382630460131, −10.00336822138179789527616639397, −9.148731688306415077836006161852, −7.80087559534190959065661153095, −7.11971322240846354786365520793, −5.41735853616393059878479849952, −4.75854258244184897850622890850, −3.76525362334355963692382899239, −1.03501633812026047468909874392, −0.07097346593938760333192102643, 2.43159794079774581304725866687, 3.66704379890857299924048121415, 5.35061740421217856988258085452, 6.15327684109702374764386057203, 7.51997185916352220455458178777, 7.71820237834816864394382375100, 9.844473006143764653049699149607, 10.40774610398041242539146154983, 11.35356602851824183423995661360, 12.24808657855263525629888930585

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