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Properties

Label 2-756-252.223-c1-0-8
Degree $2$
Conductor $756$
Sign $-0.858 + 0.513i$
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  + (0.366 + 1.36i)2-s + (−1.73 + i)4-s + (−1.5 + 0.866i)5-s + (1.73 + 2i)7-s + (−2 − 1.99i)8-s + (−1.73 − 1.73i)10-s + (0.866 + 0.5i)11-s + (−1.5 + 0.866i)13-s + (−2.09 + 3.09i)14-s + (1.99 − 3.46i)16-s + 3.46i·17-s − 6.92·19-s + (1.73 − 3i)20-s + (−0.366 + 1.36i)22-s + (−4.33 + 2.5i)23-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.5i)4-s + (−0.670 + 0.387i)5-s + (0.654 + 0.755i)7-s + (−0.707 − 0.707i)8-s + (−0.547 − 0.547i)10-s + (0.261 + 0.150i)11-s + (−0.416 + 0.240i)13-s + (−0.560 + 0.827i)14-s + (0.499 − 0.866i)16-s + 0.840i·17-s − 1.58·19-s + (0.387 − 0.670i)20-s + (−0.0780 + 0.291i)22-s + (−0.902 + 0.521i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.858 + 0.513i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.858 + 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.219950 - 0.795990i\)
\(L(\frac12)\) \(\approx\) \(0.219950 - 0.795990i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 \)
7 \( 1 + (-1.73 - 2i)T \)
good5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 + 6.92T + 19T^{2} \)
23 \( 1 + (4.33 - 2.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.33 + 7.5i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (-7.5 + 4.33i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.59 - 4.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (4.33 + 7.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.9 - 7.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 + (-2.59 - 1.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.866 + 1.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 17.3iT - 89T^{2} \)
97 \( 1 + (-4.5 - 2.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.96137145286592019500732977718, −9.753746080201278354731642597942, −8.888067995820899656611509284644, −8.043218068397099262422831708467, −7.53519115182901354943397194829, −6.36287540646109281927825085395, −5.72325639578838715584338211905, −4.45931133321390134214986435065, −3.84586497068514012862099283838, −2.23858337626806081420893150602, 0.38094486110590302123934781681, 1.85442630039743405572220415730, 3.26917324115554641724214916692, 4.39764508340137968662025273542, 4.77495606456586541525031987290, 6.15854377186646824293055521866, 7.41000380764220006565734402571, 8.344089629281822833607992227348, 8.976970774239223693327905566420, 10.17470278899608842328580493201

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