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Properties

Label 2-252-7.2-c5-0-4
Degree $2$
Conductor $252$
Sign $0.240 - 0.970i$
Analytic cond. $40.4167$
Root an. cond. $6.35741$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (41.8 − 72.4i)5-s + (28 + 126. i)7-s + (274. + 475. i)11-s − 823.·13-s + (72.7 + 126. i)17-s + (−884. + 1.53e3i)19-s + (−761. + 1.31e3i)23-s + (−1.93e3 − 3.35e3i)25-s + 741.·29-s + (1.60e3 + 2.77e3i)31-s + (1.03e4 + 3.26e3i)35-s + (1.77e3 − 3.07e3i)37-s + 6.46e3·41-s + 6.71e3·43-s + (−9.58e3 + 1.66e4i)47-s + ⋯
L(s)  = 1  + (0.748 − 1.29i)5-s + (0.215 + 0.976i)7-s + (0.684 + 1.18i)11-s − 1.35·13-s + (0.0610 + 0.105i)17-s + (−0.561 + 0.973i)19-s + (−0.300 + 0.519i)23-s + (−0.619 − 1.07i)25-s + 0.163·29-s + (0.299 + 0.518i)31-s + (1.42 + 0.450i)35-s + (0.213 − 0.369i)37-s + 0.600·41-s + 0.553·43-s + (−0.633 + 1.09i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.240 - 0.970i$
Analytic conductor: \(40.4167\)
Root analytic conductor: \(6.35741\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :5/2),\ 0.240 - 0.970i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.794661563\)
\(L(\frac12)\) \(\approx\) \(1.794661563\)
\(L(\frac{7}{2})\) 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 \( 1 + (-28 - 126. i)T \)
good5 \( 1 + (-41.8 + 72.4i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-274. - 475. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 823.T + 3.71e5T^{2} \)
17 \( 1 + (-72.7 - 126. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (884. - 1.53e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (761. - 1.31e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 741.T + 2.05e7T^{2} \)
31 \( 1 + (-1.60e3 - 2.77e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-1.77e3 + 3.07e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 6.46e3T + 1.15e8T^{2} \)
43 \( 1 - 6.71e3T + 1.47e8T^{2} \)
47 \( 1 + (9.58e3 - 1.66e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.06e4 + 1.84e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.82e4 - 3.16e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.17e4 - 3.76e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.51e3 + 4.35e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 6.31e3T + 1.80e9T^{2} \)
73 \( 1 + (-2.12e4 - 3.68e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (4.75e4 - 8.23e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 3.49e4T + 3.93e9T^{2} \)
89 \( 1 + (-5.02e4 + 8.70e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 7.67e4T + 8.58e9T^{2} \)
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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

−11.73652054933479561707462442592, −10.05896340112223669029490707365, −9.466399759222902020821501003463, −8.676258055449568184774206447868, −7.55441174983010530453534222276, −6.13839540551835219318615835179, −5.18543896619801429811725311798, −4.36257137086960936531316593597, −2.32110578485527115732593627640, −1.42658450876302465447084772900, 0.49560999593792975748657188954, 2.22979458823201348100625089627, 3.32447100098368961542900555252, 4.69342222617806804420702119122, 6.19501697777310952974941569966, 6.84455192085638075528505877148, 7.85491114509239020743882659061, 9.247661762937439198143009137567, 10.18305185873818868403953619113, 10.87235470473670328514216514858

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