Properties

Label 2-252-7.4-c11-0-14
Degree $2$
Conductor $252$
Sign $0.869 + 0.493i$
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  + (−5.57e3 − 9.66e3i)5-s + (2.49e4 + 3.68e4i)7-s + (1.96e5 − 3.39e5i)11-s − 2.28e6·13-s + (2.79e6 − 4.83e6i)17-s + (6.40e6 + 1.11e7i)19-s + (1.12e7 + 1.95e7i)23-s + (−3.78e7 + 6.55e7i)25-s − 1.65e8·29-s + (9.99e7 − 1.73e8i)31-s + (2.16e8 − 4.46e8i)35-s + (2.10e8 + 3.64e8i)37-s + 7.64e8·41-s − 2.38e8·43-s + (9.33e7 + 1.61e8i)47-s + ⋯
L(s)  = 1  + (−0.798 − 1.38i)5-s + (0.560 + 0.828i)7-s + (0.367 − 0.636i)11-s − 1.70·13-s + (0.476 − 0.826i)17-s + (0.593 + 1.02i)19-s + (0.365 + 0.633i)23-s + (−0.774 + 1.34i)25-s − 1.49·29-s + (0.626 − 1.08i)31-s + (0.697 − 1.43i)35-s + (0.498 + 0.863i)37-s + 1.03·41-s − 0.247·43-s + (0.0593 + 0.102i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.869 + 0.493i$
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: \((2,\ 252,\ (\ :11/2),\ 0.869 + 0.493i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.600846795\)
\(L(\frac12)\) \(\approx\) \(1.600846795\)
\(L(\frac{13}{2})\) 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 \( 1 + (-2.49e4 - 3.68e4i)T \)
good5 \( 1 + (5.57e3 + 9.66e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (-1.96e5 + 3.39e5i)T + (-1.42e11 - 2.47e11i)T^{2} \)
13 \( 1 + 2.28e6T + 1.79e12T^{2} \)
17 \( 1 + (-2.79e6 + 4.83e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (-6.40e6 - 1.11e7i)T + (-5.82e13 + 1.00e14i)T^{2} \)
23 \( 1 + (-1.12e7 - 1.95e7i)T + (-4.76e14 + 8.25e14i)T^{2} \)
29 \( 1 + 1.65e8T + 1.22e16T^{2} \)
31 \( 1 + (-9.99e7 + 1.73e8i)T + (-1.27e16 - 2.20e16i)T^{2} \)
37 \( 1 + (-2.10e8 - 3.64e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 - 7.64e8T + 5.50e17T^{2} \)
43 \( 1 + 2.38e8T + 9.29e17T^{2} \)
47 \( 1 + (-9.33e7 - 1.61e8i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (1.62e9 - 2.81e9i)T + (-4.63e18 - 8.02e18i)T^{2} \)
59 \( 1 + (6.84e7 - 1.18e8i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (-1.63e9 - 2.82e9i)T + (-2.17e19 + 3.76e19i)T^{2} \)
67 \( 1 + (7.66e9 - 1.32e10i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 + 1.53e10T + 2.31e20T^{2} \)
73 \( 1 + (-1.23e10 + 2.13e10i)T + (-1.56e20 - 2.71e20i)T^{2} \)
79 \( 1 + (-1.05e10 - 1.83e10i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 - 5.62e10T + 1.28e21T^{2} \)
89 \( 1 + (-1.87e10 - 3.23e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 + 7.72e10T + 7.15e21T^{2} \)
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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

−9.646182576035307417841449957164, −9.148938345012945923087740095424, −7.987039235780467872561681284775, −7.55198573291142412752088512976, −5.73336217183901594270897817573, −5.07238257673916246571972915638, −4.16405635131337691730257151487, −2.83833234371426986665270661523, −1.51361466176660098069319843229, −0.54572271643007910276577895571, 0.53195894322323973344126076514, 2.02073769831317100606316175060, 3.07521832984523932125471460852, 4.09022099878679852317580547314, 5.01710725403319657875517520898, 6.66364963461499641069488938887, 7.32072489244197583153476529741, 7.83902542653661929498118617237, 9.413362094401524638211401947021, 10.36619166144081929056579764791

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