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Properties

Label 2-756-21.2-c2-0-8
Degree $2$
Conductor $756$
Sign $0.607 - 0.794i$
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  + (5.22 + 3.01i)5-s + (2.27 + 6.62i)7-s + (8.17 − 4.71i)11-s + 13.3·13-s + (−9.36 + 5.40i)17-s + (2.20 − 3.81i)19-s + (17.2 + 9.95i)23-s + (5.67 + 9.83i)25-s + 10.6i·29-s + (−17.4 − 30.3i)31-s + (−8.09 + 41.4i)35-s + (−0.378 + 0.655i)37-s + 32.4i·41-s − 44.3·43-s + (70.3 + 40.6i)47-s + ⋯
L(s)  = 1  + (1.04 + 0.602i)5-s + (0.324 + 0.945i)7-s + (0.742 − 0.428i)11-s + 1.02·13-s + (−0.551 + 0.318i)17-s + (0.115 − 0.200i)19-s + (0.749 + 0.432i)23-s + (0.227 + 0.393i)25-s + 0.368i·29-s + (−0.564 − 0.977i)31-s + (−0.231 + 1.18i)35-s + (−0.0102 + 0.0177i)37-s + 0.790i·41-s − 1.03·43-s + (1.49 + 0.863i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.607 - 0.794i$
Analytic conductor: \(20.5995\)
Root analytic conductor: \(4.53866\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1),\ 0.607 - 0.794i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.518163005\)
\(L(\frac12)\) \(\approx\) \(2.518163005\)
\(L(2)\) 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 \( 1 + (-2.27 - 6.62i)T \)
good5 \( 1 + (-5.22 - 3.01i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-8.17 + 4.71i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 13.3T + 169T^{2} \)
17 \( 1 + (9.36 - 5.40i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-2.20 + 3.81i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-17.2 - 9.95i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 10.6iT - 841T^{2} \)
31 \( 1 + (17.4 + 30.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (0.378 - 0.655i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 32.4iT - 1.68e3T^{2} \)
43 \( 1 + 44.3T + 1.84e3T^{2} \)
47 \( 1 + (-70.3 - 40.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (17.0 - 9.87i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-38.9 + 22.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (20.5 - 35.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (45.7 + 79.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 11.2iT - 5.04e3T^{2} \)
73 \( 1 + (-10.2 - 17.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-57.5 + 99.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 35.0iT - 6.88e3T^{2} \)
89 \( 1 + (-45.2 - 26.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 182.T + 9.40e3T^{2} \)
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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

−10.30817943909287376532801628014, −9.166098884561919907005943459566, −8.911528970182799448837236122639, −7.70389893348012242424426619133, −6.37880123027340343646152107105, −6.09548022578678313063766055730, −5.03669929883611999862210475262, −3.63794482998280318510402004702, −2.50953999388081823079282149173, −1.44369415541577494764233150769, 0.990180422018044978877612690957, 1.93099855349301778703553506385, 3.60197725221195807950600611655, 4.58852130401421023750736598633, 5.51297439486645159593781559842, 6.54918564138296473995070332630, 7.26884750084388895247848752655, 8.564222566942082395413576187518, 9.098356269332476515326400788034, 10.03359507296862758215601270264

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