Properties

Label 2-252-63.5-c3-0-21
Degree $2$
Conductor $252$
Sign $-0.795 - 0.606i$
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  + (−5.18 + 0.406i)3-s + (−8.68 − 15.0i)5-s + (6.03 − 17.5i)7-s + (26.6 − 4.21i)9-s + (−9.11 − 5.26i)11-s + (−11.0 − 6.35i)13-s + (51.1 + 74.4i)15-s + (−28.9 − 50.2i)17-s + (56.6 + 32.7i)19-s + (−24.1 + 93.1i)21-s + (−31.2 + 18.0i)23-s + (−88.3 + 153. i)25-s + (−136. + 32.6i)27-s + (−230. + 133. i)29-s − 162. i·31-s + ⋯
L(s)  = 1  + (−0.996 + 0.0782i)3-s + (−0.776 − 1.34i)5-s + (0.326 − 0.945i)7-s + (0.987 − 0.156i)9-s + (−0.249 − 0.144i)11-s + (−0.235 − 0.135i)13-s + (0.879 + 1.28i)15-s + (−0.413 − 0.716i)17-s + (0.684 + 0.395i)19-s + (−0.251 + 0.967i)21-s + (−0.283 + 0.163i)23-s + (−0.707 + 1.22i)25-s + (−0.972 + 0.232i)27-s + (−1.47 + 0.852i)29-s − 0.942i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.795 - 0.606i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.795 - 0.606i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2754383910\)
\(L(\frac12)\) \(\approx\) \(0.2754383910\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.18 - 0.406i)T \)
7 \( 1 + (-6.03 + 17.5i)T \)
good5 \( 1 + (8.68 + 15.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (9.11 + 5.26i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (11.0 + 6.35i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (28.9 + 50.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-56.6 - 32.7i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (31.2 - 18.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (230. - 133. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 162. iT - 2.97e4T^{2} \)
37 \( 1 + (59.1 - 102. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-33.7 + 58.5i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-136. - 235. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 129.T + 1.03e5T^{2} \)
53 \( 1 + (39.3 - 22.7i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 427.T + 2.05e5T^{2} \)
61 \( 1 - 896. iT - 2.26e5T^{2} \)
67 \( 1 + 72.6T + 3.00e5T^{2} \)
71 \( 1 - 664. iT - 3.57e5T^{2} \)
73 \( 1 + (363. - 209. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 1.19e3T + 4.93e5T^{2} \)
83 \( 1 + (711. + 1.23e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-759. + 1.31e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.01e3 - 584. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.29250563021190425166579584762, −10.16742450993575190531623364853, −9.155860841417701794375959728650, −7.86705395569443963894846967842, −7.19206528507893763302972627093, −5.60187969207912097112811336570, −4.75075260258642669431304727587, −3.88959175934149031789698253545, −1.23190164244639347809400102129, −0.13816179796873030437285153524, 2.18592181713305423608381682013, 3.72270526919838285725668693234, 5.10801765507481778653268829313, 6.20351039642835367594806523849, 7.12020547160353696973975141746, 8.003621945524256724952981665117, 9.478787423011018785680182154864, 10.63935150317572492639524857869, 11.23051417869378315672531830835, 11.93835371615553173324210419861

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