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Properties

Label 2-756-9.4-c1-0-2
Degree $2$
Conductor $756$
Sign $0.991 - 0.128i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.23 − 2.13i)5-s + (0.5 + 0.866i)7-s + (2.32 + 4.02i)11-s + (−3.55 + 6.15i)13-s + 4.51·17-s + 4.32·19-s + (2.93 − 5.08i)23-s + (−0.527 − 0.912i)25-s + (−3.48 − 6.04i)29-s + (3.69 − 6.39i)31-s + 2.46·35-s − 0.726·37-s + (0.136 − 0.236i)41-s + (2.41 + 4.18i)43-s + (1.83 + 3.18i)47-s + ⋯
L(s)  = 1  + (0.550 − 0.952i)5-s + (0.188 + 0.327i)7-s + (0.700 + 1.21i)11-s + (−0.985 + 1.70i)13-s + 1.09·17-s + 0.992·19-s + (0.611 − 1.05i)23-s + (−0.105 − 0.182i)25-s + (−0.647 − 1.12i)29-s + (0.662 − 1.14i)31-s + 0.415·35-s − 0.119·37-s + (0.0213 − 0.0369i)41-s + (0.368 + 0.638i)43-s + (0.267 + 0.463i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.991 - 0.128i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.991 - 0.128i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78636 + 0.115343i\)
\(L(\frac12)\) \(\approx\) \(1.78636 + 0.115343i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-1.23 + 2.13i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.32 - 4.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.55 - 6.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.51T + 17T^{2} \)
19 \( 1 - 4.32T + 19T^{2} \)
23 \( 1 + (-2.93 + 5.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.48 + 6.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.69 + 6.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.726T + 37T^{2} \)
41 \( 1 + (-0.136 + 0.236i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.83 - 3.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.05T + 53T^{2} \)
59 \( 1 + (-4.56 + 7.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.90 - 11.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.663 + 1.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + 4.32T + 73T^{2} \)
79 \( 1 + (3.21 + 5.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.742 - 1.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.83T + 89T^{2} \)
97 \( 1 + (-0.246 - 0.426i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.914815566210789384195134755279, −9.552026435431906320660366567632, −8.928098947220523875391857130569, −7.72587829993190728758704632380, −6.92689631573236672541201299348, −5.84520855778755482136443900236, −4.82595423664910743145180421057, −4.24079307093870095856474568340, −2.42723228903060224390699962440, −1.38284535622785182245152206167, 1.12859671494676234831687650753, 2.98331655454205316433696737050, 3.40300661289872818950601619119, 5.24795812566315373494790228231, 5.73987129356478244018004058535, 6.96753043642573184459411007801, 7.58042181518562402398983495248, 8.596944568385859470432232738890, 9.706001783510572742139133314222, 10.31895669784349060948171191769

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