Properties

Label 2-252-7.4-c11-0-26
Degree $2$
Conductor $252$
Sign $0.674 - 0.738i$
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  + (6.71e3 + 1.16e4i)5-s + (4.14e4 − 1.61e4i)7-s + (4.25e5 − 7.37e5i)11-s + 2.13e6·13-s + (−1.28e6 + 2.21e6i)17-s + (8.87e6 + 1.53e7i)19-s + (1.81e7 + 3.14e7i)23-s + (−6.58e7 + 1.14e8i)25-s − 5.71e5·29-s + (1.07e8 − 1.86e8i)31-s + (4.65e8 + 3.74e8i)35-s + (−9.87e7 − 1.71e8i)37-s + 2.72e8·41-s + 1.11e9·43-s + (−3.98e8 − 6.90e8i)47-s + ⋯
L(s)  = 1  + (0.961 + 1.66i)5-s + (0.932 − 0.362i)7-s + (0.796 − 1.38i)11-s + 1.59·13-s + (−0.218 + 0.379i)17-s + (0.822 + 1.42i)19-s + (0.588 + 1.01i)23-s + (−1.34 + 2.33i)25-s − 0.00517·29-s + (0.676 − 1.17i)31-s + (1.49 + 1.20i)35-s + (−0.234 − 0.405i)37-s + 0.367·41-s + 1.15·43-s + (−0.253 − 0.439i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.674 - 0.738i$
Analytic conductor: \(193.622\)
Root analytic conductor: \(13.9148\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :11/2),\ 0.674 - 0.738i)\)

Particular Values

\(L(6)\) \(\approx\) \(4.427106190\)
\(L(\frac12)\) \(\approx\) \(4.427106190\)
\(L(\frac{13}{2})\) 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 \( 1 + (-4.14e4 + 1.61e4i)T \)
good5 \( 1 + (-6.71e3 - 1.16e4i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (-4.25e5 + 7.37e5i)T + (-1.42e11 - 2.47e11i)T^{2} \)
13 \( 1 - 2.13e6T + 1.79e12T^{2} \)
17 \( 1 + (1.28e6 - 2.21e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (-8.87e6 - 1.53e7i)T + (-5.82e13 + 1.00e14i)T^{2} \)
23 \( 1 + (-1.81e7 - 3.14e7i)T + (-4.76e14 + 8.25e14i)T^{2} \)
29 \( 1 + 5.71e5T + 1.22e16T^{2} \)
31 \( 1 + (-1.07e8 + 1.86e8i)T + (-1.27e16 - 2.20e16i)T^{2} \)
37 \( 1 + (9.87e7 + 1.71e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 - 2.72e8T + 5.50e17T^{2} \)
43 \( 1 - 1.11e9T + 9.29e17T^{2} \)
47 \( 1 + (3.98e8 + 6.90e8i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (-6.76e8 + 1.17e9i)T + (-4.63e18 - 8.02e18i)T^{2} \)
59 \( 1 + (3.81e8 - 6.60e8i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (2.04e9 + 3.53e9i)T + (-2.17e19 + 3.76e19i)T^{2} \)
67 \( 1 + (-1.08e9 + 1.88e9i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 + 1.82e10T + 2.31e20T^{2} \)
73 \( 1 + (-1.03e10 + 1.78e10i)T + (-1.56e20 - 2.71e20i)T^{2} \)
79 \( 1 + (1.28e10 + 2.22e10i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 + 1.36e10T + 1.28e21T^{2} \)
89 \( 1 + (5.18e9 + 8.98e9i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 + 1.14e11T + 7.15e21T^{2} \)
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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

−10.45931559405104661166000746763, −9.372081130343989798779320405904, −8.257839243884824484354453519933, −7.26087901952886165470604998913, −6.03276386664777985873817711201, −5.83434026703738793234567285898, −3.82438618651926512902056832307, −3.23442189019959013395617368031, −1.82831489027591022237877783284, −1.07517874771252213993681255336, 1.00644817602059270456660210377, 1.33580622600207863936060827527, 2.47615195156310258939024776306, 4.41366787700027082055681333614, 4.83765664047599849903658057940, 5.85647110025522804938180847494, 7.00298748939348184229606389623, 8.486328885722477306222478934705, 8.921556162218434248848121427079, 9.685706646452384241109266480955

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