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Properties

Label 2-756-252.223-c1-0-12
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  + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (−1.5 + 0.866i)5-s + (−1.73 − 2i)7-s + (−1.99 + 2i)8-s + (1.73 − 1.73i)10-s + (−0.866 − 0.5i)11-s + (−1.5 + 0.866i)13-s + (3.09 + 2.09i)14-s + (1.99 − 3.46i)16-s + 3.46i·17-s + 6.92·19-s + (−1.73 + 3i)20-s + (1.36 + 0.366i)22-s + (4.33 − 2.5i)23-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)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 + (0.827 + 0.560i)14-s + (0.499 − 0.866i)16-s + 0.840i·17-s + 1.58·19-s + (−0.387 + 0.670i)20-s + (0.291 + 0.0780i)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

\(L(1)\) \(\approx\) \(0.715717 + 0.197769i\)
\(L(\frac12)\) \(\approx\) \(0.715717 + 0.197769i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
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

−10.30841346854580831750411020366, −9.643438616239190658344097000893, −8.676418875493912069475054836614, −7.72059656714632447957580138596, −7.15113471480405765249119608047, −6.38940493554191767021458088283, −5.19044634702820855042486595199, −3.73843150086036358401906539328, −2.72892097497309580346862467534, −0.906404950604827799280188765987, 0.73911426873657444621242409896, 2.55522509767766928135146504693, 3.36210505924807439539173357009, 4.87279657156461920240910539850, 5.97518713117145819464418938427, 7.13675092744330821070774135149, 7.76436872309544730512725963293, 8.647527138393197581985148602935, 9.562950314809820624752644363187, 9.882650692042244486779363963298

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