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Properties

Label 2-252-21.5-c3-0-5
Degree $2$
Conductor $252$
Sign $0.468 + 0.883i$
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  + (10.1 − 17.5i)5-s + (12.0 + 14.0i)7-s + (15.4 − 8.92i)11-s − 33.1i·13-s + (22.9 + 39.6i)17-s + (−35.0 − 20.2i)19-s + (69.7 + 40.2i)23-s + (−142. − 246. i)25-s − 233. i·29-s + (−195. + 112. i)31-s + (368. − 68.3i)35-s + (135. − 234. i)37-s + 154.·41-s + 367.·43-s + (263. − 457. i)47-s + ⋯
L(s)  = 1  + (0.905 − 1.56i)5-s + (0.649 + 0.760i)7-s + (0.423 − 0.244i)11-s − 0.707i·13-s + (0.326 + 0.566i)17-s + (−0.422 − 0.244i)19-s + (0.632 + 0.365i)23-s + (−1.14 − 1.97i)25-s − 1.49i·29-s + (−1.13 + 0.653i)31-s + (1.78 − 0.329i)35-s + (0.601 − 1.04i)37-s + 0.588·41-s + 1.30·43-s + (0.818 − 1.41i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.468 + 0.883i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.468 + 0.883i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.250550665\)
\(L(\frac12)\) \(\approx\) \(2.250550665\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 + (-12.0 - 14.0i)T \)
good5 \( 1 + (-10.1 + 17.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-15.4 + 8.92i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 33.1iT - 2.19e3T^{2} \)
17 \( 1 + (-22.9 - 39.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (35.0 + 20.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-69.7 - 40.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 233. iT - 2.43e4T^{2} \)
31 \( 1 + (195. - 112. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-135. + 234. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 154.T + 6.89e4T^{2} \)
43 \( 1 - 367.T + 7.95e4T^{2} \)
47 \( 1 + (-263. + 457. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (78.8 - 45.5i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-312. - 541. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (78.8 + 45.5i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (431. + 747. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 303. iT - 3.57e5T^{2} \)
73 \( 1 + (999. - 576. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-3.48 + 6.02i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 815.T + 5.71e5T^{2} \)
89 \( 1 + (155. - 269. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.83e3iT - 9.12e5T^{2} \)
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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

−11.59107086150966235351605179249, −10.40154892438058539385030064280, −9.190482178134386119843191469242, −8.783777909866617676704039098959, −7.73947845471508530935107094691, −5.90580370184819947927334661914, −5.42980053045811695469525169987, −4.23613319412373180125020763974, −2.22996470999556180095862059994, −0.986610211091218122616689936139, 1.62825289044305193895389147982, 2.95840187237013479615099002153, 4.35146583977549334363703481912, 5.83426999493023161600740407462, 6.89735233830048229269928235813, 7.45875376982559736813466316079, 9.067899779272498311017793264765, 9.995954976405570946658789160442, 10.86560797408018812748885574473, 11.38029606765932707995845084022

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