Properties

Degree $2$
Conductor $252$
Sign $0.991 + 0.126i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 − 0.866i)7-s + 5·13-s + (0.5 − 0.866i)19-s + (2.5 + 4.33i)25-s + (−5.5 − 9.52i)31-s + (−5.5 + 9.52i)37-s − 13·43-s + (5.5 − 4.33i)49-s + (−7 + 12.1i)61-s + (−2.5 − 4.33i)67-s + (−8.5 − 14.7i)73-s + (−8.5 + 14.7i)79-s + (12.5 − 4.33i)91-s + 14·97-s + (6.5 − 11.2i)103-s + ⋯
L(s)  = 1  + (0.944 − 0.327i)7-s + 1.38·13-s + (0.114 − 0.198i)19-s + (0.5 + 0.866i)25-s + (−0.987 − 1.71i)31-s + (−0.904 + 1.56i)37-s − 1.98·43-s + (0.785 − 0.618i)49-s + (−0.896 + 1.55i)61-s + (−0.305 − 0.529i)67-s + (−0.994 − 1.72i)73-s + (−0.956 + 1.65i)79-s + (1.31 − 0.453i)91-s + 1.42·97-s + (0.640 − 1.10i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.991 + 0.126i$
Motivic weight: \(1\)
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39091 - 0.0882674i\)
\(L(\frac12)\) \(\approx\) \(1.39091 - 0.0882674i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (5.5 + 9.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 13T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (8.5 + 14.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−11.76781945541260148277579085538, −11.16649553244544186493239264889, −10.26770479503104807237374766039, −8.989685743507978045566270894294, −8.172325327717420668150147378323, −7.13029545854197214861403933751, −5.89035506414127979385176340765, −4.71978140179157600387267003743, −3.46622239353313215930627619790, −1.54297146000919957214995670622, 1.67259348408527152003717601706, 3.46639878577192253208900107072, 4.84025151217890051943393771530, 5.90699610136419423916435487577, 7.12169118176407111083321705790, 8.372007332579292663194765642537, 8.891602898263775853725812511253, 10.37132101945693126350887489407, 11.08157249016867279069507421164, 11.99929450501712945365765238435

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