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Properties

Label 2-756-63.41-c1-0-4
Degree $2$
Conductor $756$
Sign $0.999 + 0.0285i$
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  + (1.21 − 2.10i)5-s + (1.57 + 2.12i)7-s + (−2.09 + 1.21i)11-s + (4.73 + 2.73i)13-s + 2.58·17-s − 0.402i·19-s + (−3.06 − 1.77i)23-s + (−0.440 − 0.762i)25-s + (6.31 − 3.64i)29-s + (3.63 + 2.10i)31-s + (6.37 − 0.722i)35-s − 3.19·37-s + (4.03 − 6.99i)41-s + (−4.22 − 7.31i)43-s + (−2.25 − 3.91i)47-s + ⋯
L(s)  = 1  + (0.542 − 0.939i)5-s + (0.594 + 0.804i)7-s + (−0.632 + 0.365i)11-s + (1.31 + 0.758i)13-s + 0.625·17-s − 0.0923i·19-s + (−0.639 − 0.369i)23-s + (−0.0880 − 0.152i)25-s + (1.17 − 0.677i)29-s + (0.653 + 0.377i)31-s + (1.07 − 0.122i)35-s − 0.525·37-s + (0.630 − 1.09i)41-s + (−0.644 − 1.11i)43-s + (−0.329 − 0.570i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.999 + 0.0285i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.999 + 0.0285i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85010 - 0.0264138i\)
\(L(\frac12)\) \(\approx\) \(1.85010 - 0.0264138i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 - 2.12i)T \)
good5 \( 1 + (-1.21 + 2.10i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.09 - 1.21i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.73 - 2.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.58T + 17T^{2} \)
19 \( 1 + 0.402iT - 19T^{2} \)
23 \( 1 + (3.06 + 1.77i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.31 + 3.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.63 - 2.10i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.19T + 37T^{2} \)
41 \( 1 + (-4.03 + 6.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.22 + 7.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.25 + 3.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 14.0iT - 53T^{2} \)
59 \( 1 + (0.0779 - 0.134i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.2 + 5.90i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.53 + 4.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.73iT - 71T^{2} \)
73 \( 1 + 8.80iT - 73T^{2} \)
79 \( 1 + (-5.66 - 9.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.50 - 13.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (4.97 - 2.87i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.27680169643590610542740369413, −9.368574228213761229304619547107, −8.573311450971396551698538012659, −8.117352714389880174989182884891, −6.72402286483027628939443832571, −5.71686548786564394249836445680, −5.07604508771360114263950847833, −4.03600092289739258622730186679, −2.43528498912101745097954362825, −1.33210504397716376657989940239, 1.22916886486803121668358389847, 2.79539046975826208752686411059, 3.69839497461931787473241692677, 5.02220786646911315568803473460, 6.03248960856052817323936468025, 6.75499596364940344174645937091, 7.941524237285889414817821324920, 8.323993725225128974353059502961, 9.837818734670140018805983709168, 10.39358210889538313588448508104

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