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Properties

Label 2-252-9.4-c3-0-3
Degree $2$
Conductor $252$
Sign $-0.995 + 0.0922i$
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  + (−1.54 + 4.95i)3-s + (3.85 − 6.68i)5-s + (−3.5 − 6.06i)7-s + (−22.1 − 15.3i)9-s + (20.9 + 36.2i)11-s + (−19.8 + 34.3i)13-s + (27.1 + 29.4i)15-s − 68.0·17-s − 104.·19-s + (35.4 − 7.96i)21-s + (−72.9 + 126. i)23-s + (32.7 + 56.6i)25-s + (110. − 86.2i)27-s + (−145. − 252. i)29-s + (133. − 231. i)31-s + ⋯
L(s)  = 1  + (−0.298 + 0.954i)3-s + (0.345 − 0.597i)5-s + (−0.188 − 0.327i)7-s + (−0.822 − 0.569i)9-s + (0.574 + 0.994i)11-s + (−0.422 + 0.732i)13-s + (0.467 + 0.507i)15-s − 0.970·17-s − 1.25·19-s + (0.368 − 0.0827i)21-s + (−0.661 + 1.14i)23-s + (0.261 + 0.453i)25-s + (0.788 − 0.614i)27-s + (−0.932 − 1.61i)29-s + (0.773 − 1.34i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.995 + 0.0922i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.995 + 0.0922i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3903671352\)
\(L(\frac12)\) \(\approx\) \(0.3903671352\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.54 - 4.95i)T \)
7 \( 1 + (3.5 + 6.06i)T \)
good5 \( 1 + (-3.85 + 6.68i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-20.9 - 36.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (19.8 - 34.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 68.0T + 4.91e3T^{2} \)
19 \( 1 + 104.T + 6.85e3T^{2} \)
23 \( 1 + (72.9 - 126. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (145. + 252. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-133. + 231. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 341.T + 5.06e4T^{2} \)
41 \( 1 + (116. - 201. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-20.6 - 35.8i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (120. + 209. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 505.T + 1.48e5T^{2} \)
59 \( 1 + (90.6 - 156. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-339. - 587. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (443. - 767. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 594.T + 3.57e5T^{2} \)
73 \( 1 - 704.T + 3.89e5T^{2} \)
79 \( 1 + (-287. - 498. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (170. + 294. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.42e3T + 7.04e5T^{2} \)
97 \( 1 + (-76.5 - 132. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.89595919829718256146908229502, −11.18931284103441157548296640961, −9.860443733439734972768155993454, −9.537814447560481701996103678390, −8.474040230752313258692266028613, −6.97224816456531173641427492371, −5.91888778206756997210303316158, −4.62203963175076122254257613485, −4.00048499110423365544963284597, −2.00262814905749252876842253413, 0.14832214323277938119257095144, 1.98553953958511366318067270576, 3.18930303415710501669306355703, 5.09463559394831999248909406931, 6.39585563794093672182671387432, 6.70329137683538243124687440500, 8.226272805914296760985446981215, 8.902669245072264789629796948200, 10.54538779655609822260773053575, 10.96894197002801509140107591605

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