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Properties

Label 2-756-7.3-c2-0-11
Degree $2$
Conductor $756$
Sign $0.999 + 0.0165i$
Analytic cond. $20.5995$
Root an. cond. $4.53866$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 6.92i)7-s − 25.9i·13-s + (24 − 13.8i)19-s + (−12.5 + 21.6i)25-s + (52.5 + 30.3i)31-s + (23.5 + 40.7i)37-s + 61·43-s + (−46.9 − 13.8i)49-s + (97.5 − 56.2i)61-s + (54.5 − 94.3i)67-s + (120 + 69.2i)73-s + (−65.5 − 113. i)79-s + (180 + 25.9i)91-s + 95.2i·97-s + (−175.5 + 101. i)103-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)7-s − 1.99i·13-s + (1.26 − 0.729i)19-s + (−0.5 + 0.866i)25-s + (1.69 + 0.977i)31-s + (0.635 + 1.10i)37-s + 1.41·43-s + (−0.959 − 0.282i)49-s + (1.59 − 0.922i)61-s + (0.813 − 1.40i)67-s + (1.64 + 0.949i)73-s + (−0.829 − 1.43i)79-s + (1.97 + 0.285i)91-s + 0.982i·97-s + (−1.70 + 0.983i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0165i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.999 + 0.0165i$
Analytic conductor: \(20.5995\)
Root analytic conductor: \(4.53866\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1),\ 0.999 + 0.0165i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.835692450\)
\(L(\frac12)\) \(\approx\) \(1.835692450\)
\(L(2)\) 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 \( 1 + (1 - 6.92i)T \)
good5 \( 1 + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 25.9iT - 169T^{2} \)
17 \( 1 + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-24 + 13.8i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 841T^{2} \)
31 \( 1 + (-52.5 - 30.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-23.5 - 40.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 61T + 1.84e3T^{2} \)
47 \( 1 + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-97.5 + 56.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-54.5 + 94.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 5.04e3T^{2} \)
73 \( 1 + (-120 - 69.2i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (65.5 + 113. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 95.2iT - 9.40e3T^{2} \)
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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

−10.02095862710318444675731854851, −9.382951432796655410231656925479, −8.334178410316347508708379663016, −7.73729636261113171769364985928, −6.55411938175752420868010541701, −5.55378113429104500795746660793, −4.99234295592989487063688917126, −3.33796403137262253926594716758, −2.63588596354056178943296316536, −0.887739736429597782747576954097, 0.958653013732935254240511749481, 2.35703640745463054226798789130, 3.90144159699070434728390289921, 4.40636601513099559497744753848, 5.81106448956747275137085565643, 6.73697380326256297213565021866, 7.46542230588668859006428427653, 8.380467038711262573335464690038, 9.572419445298507969830384001570, 9.901596327104109752522462137889

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