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Properties

Label 2-252-7.2-c11-0-36
Degree $2$
Conductor $252$
Sign $-0.336 - 0.941i$
Analytic cond. $193.622$
Root an. cond. $13.9148$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.53e3 − 9.59e3i)5-s + (−3.20e4 − 3.08e4i)7-s + (−4.51e5 − 7.82e5i)11-s − 7.61e5·13-s + (−2.39e6 − 4.14e6i)17-s + (−2.25e6 + 3.90e6i)19-s + (2.38e7 − 4.13e7i)23-s + (−3.69e7 − 6.40e7i)25-s − 1.65e8·29-s + (−2.95e7 − 5.11e7i)31-s + (−4.73e8 + 1.36e8i)35-s + (1.08e8 − 1.87e8i)37-s + 5.52e8·41-s − 1.04e9·43-s + (6.93e7 − 1.20e8i)47-s + ⋯
L(s)  = 1  + (0.792 − 1.37i)5-s + (−0.720 − 0.693i)7-s + (−0.845 − 1.46i)11-s − 0.568·13-s + (−0.408 − 0.707i)17-s + (−0.208 + 0.361i)19-s + (0.773 − 1.34i)23-s + (−0.757 − 1.31i)25-s − 1.49·29-s + (−0.185 − 0.321i)31-s + (−1.52 + 0.438i)35-s + (0.257 − 0.445i)37-s + 0.744·41-s − 1.07·43-s + (0.0440 − 0.0763i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.336 - 0.941i$
Analytic conductor: \(193.622\)
Root analytic conductor: \(13.9148\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :11/2),\ -0.336 - 0.941i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.176269309\)
\(L(\frac12)\) \(\approx\) \(1.176269309\)
\(L(\frac{13}{2})\) 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 \( 1 + (3.20e4 + 3.08e4i)T \)
good5 \( 1 + (-5.53e3 + 9.59e3i)T + (-2.44e7 - 4.22e7i)T^{2} \)
11 \( 1 + (4.51e5 + 7.82e5i)T + (-1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 + 7.61e5T + 1.79e12T^{2} \)
17 \( 1 + (2.39e6 + 4.14e6i)T + (-1.71e13 + 2.96e13i)T^{2} \)
19 \( 1 + (2.25e6 - 3.90e6i)T + (-5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (-2.38e7 + 4.13e7i)T + (-4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 + 1.65e8T + 1.22e16T^{2} \)
31 \( 1 + (2.95e7 + 5.11e7i)T + (-1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (-1.08e8 + 1.87e8i)T + (-8.89e16 - 1.54e17i)T^{2} \)
41 \( 1 - 5.52e8T + 5.50e17T^{2} \)
43 \( 1 + 1.04e9T + 9.29e17T^{2} \)
47 \( 1 + (-6.93e7 + 1.20e8i)T + (-1.23e18 - 2.14e18i)T^{2} \)
53 \( 1 + (2.34e9 + 4.05e9i)T + (-4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (-4.49e9 - 7.78e9i)T + (-1.50e19 + 2.61e19i)T^{2} \)
61 \( 1 + (1.08e8 - 1.87e8i)T + (-2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (8.64e9 + 1.49e10i)T + (-6.10e19 + 1.05e20i)T^{2} \)
71 \( 1 - 5.28e9T + 2.31e20T^{2} \)
73 \( 1 + (5.55e9 + 9.61e9i)T + (-1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (-2.14e9 + 3.71e9i)T + (-3.73e20 - 6.47e20i)T^{2} \)
83 \( 1 - 4.61e10T + 1.28e21T^{2} \)
89 \( 1 + (-8.53e9 + 1.47e10i)T + (-1.38e21 - 2.40e21i)T^{2} \)
97 \( 1 - 1.57e11T + 7.15e21T^{2} \)
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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

−9.350083070467700492764115125071, −8.675205886239377810401377372799, −7.59320650735624385958957534869, −6.30975626108784669247343206796, −5.43315127876946217803744589917, −4.56467507118529143723376060039, −3.25143028475307408464414860597, −2.08412730829547004033830322807, −0.67737367748037752494171566098, −0.28188280329616078773459696754, 1.87266089150883408800961461179, 2.47825707770375499585747206707, 3.43080275841008032974127127163, 4.99796176120512402630098935810, 5.96882362392777332423450466318, 6.89909746934997563108833372222, 7.57427946062737165056632323997, 9.208610481924470611756448275500, 9.842435459622547355521754298744, 10.57677274576322338481788112412

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