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Properties

Label 2-756-7.4-c1-0-9
Degree $2$
Conductor $756$
Sign $-0.991 + 0.126i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 1.73i)7-s − 7·13-s + (−4 − 6.92i)19-s + (2.5 − 4.33i)25-s + (−5.5 + 9.52i)31-s + (0.5 + 0.866i)37-s − 13·43-s + (1.00 − 6.92i)49-s + (0.5 + 0.866i)61-s + (−5.5 + 9.52i)67-s + (5 − 8.66i)73-s + (6.5 + 11.2i)79-s + (14 − 12.1i)91-s − 19·97-s + (3.5 + 6.06i)103-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)7-s − 1.94·13-s + (−0.917 − 1.58i)19-s + (0.5 − 0.866i)25-s + (−0.987 + 1.71i)31-s + (0.0821 + 0.142i)37-s − 1.98·43-s + (0.142 − 0.989i)49-s + (0.0640 + 0.110i)61-s + (−0.671 + 1.16i)67-s + (0.585 − 1.01i)73-s + (0.731 + 1.26i)79-s + (1.46 − 1.27i)91-s − 1.92·97-s + (0.344 + 0.597i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 7T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 19T + 97T^{2} \)
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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

−9.887322470137963680809190392534, −9.120512558456089903221012837588, −8.381092038084311284602853670839, −7.06476342203788797954215674069, −6.64877547330098715349704762108, −5.30836994320640864202856382895, −4.62216698753173184103127266406, −3.09513733736063924369718113462, −2.26555039051722170913284208592, 0, 2.01116051270762943663652030735, 3.33131925600993062102943193966, 4.32295978299945350176159014135, 5.41228734321090273550459107508, 6.46392234880406541068158958085, 7.32659321742139427561074640349, 8.001563899498960574820874494160, 9.291735066432182844123465416347, 9.904687386031125116896321677541

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