Properties

Label 2-756-12.11-c1-0-36
Degree $2$
Conductor $756$
Sign $-0.809 + 0.587i$
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  + (−0.436 − 1.34i)2-s + (−1.61 + 1.17i)4-s − 3.88i·5-s + i·7-s + (2.28 + 1.66i)8-s + (−5.21 + 1.69i)10-s + 5.38·11-s + 3.79·13-s + (1.34 − 0.436i)14-s + (1.24 − 3.80i)16-s − 4.17i·17-s − 3.62i·19-s + (4.55 + 6.28i)20-s + (−2.35 − 7.24i)22-s − 2.83·23-s + ⋯
L(s)  = 1  + (−0.308 − 0.951i)2-s + (−0.809 + 0.587i)4-s − 1.73i·5-s + 0.377i·7-s + (0.808 + 0.588i)8-s + (−1.65 + 0.535i)10-s + 1.62·11-s + 1.05·13-s + (0.359 − 0.116i)14-s + (0.310 − 0.950i)16-s − 1.01i·17-s − 0.831i·19-s + (1.01 + 1.40i)20-s + (−0.501 − 1.54i)22-s − 0.590·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.809 + 0.587i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.809 + 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.392445 - 1.20915i\)
\(L(\frac12)\) \(\approx\) \(0.392445 - 1.20915i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.436 + 1.34i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 + 3.88iT - 5T^{2} \)
11 \( 1 - 5.38T + 11T^{2} \)
13 \( 1 - 3.79T + 13T^{2} \)
17 \( 1 + 4.17iT - 17T^{2} \)
19 \( 1 + 3.62iT - 19T^{2} \)
23 \( 1 + 2.83T + 23T^{2} \)
29 \( 1 - 4.11iT - 29T^{2} \)
31 \( 1 + 1.98iT - 31T^{2} \)
37 \( 1 + 1.32T + 37T^{2} \)
41 \( 1 + 4.35iT - 41T^{2} \)
43 \( 1 - 10.8iT - 43T^{2} \)
47 \( 1 + 3.47T + 47T^{2} \)
53 \( 1 + 8.37iT - 53T^{2} \)
59 \( 1 - 0.310T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + 10.2iT - 67T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 + 4.88T + 73T^{2} \)
79 \( 1 - 2.10iT - 79T^{2} \)
83 \( 1 - 9.51T + 83T^{2} \)
89 \( 1 + 8.60iT - 89T^{2} \)
97 \( 1 - 19.0T + 97T^{2} \)
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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

−9.688689496737784216572400339261, −9.112160736687781774610274870186, −8.758128676327543915660459552090, −7.82870241231046259114725208530, −6.39185977635889230795562324968, −5.15373574040748811025187393036, −4.42135821905733059991752916096, −3.45867737682862280545824352172, −1.77726285557656279513150470235, −0.824112126750273088368710012903, 1.60786616416227843933681744032, 3.58067668238953959426407920944, 4.07893982084686301696633991777, 5.98508366827969117212622939398, 6.30991160242919151283555318024, 7.08008369325761637460219424736, 7.957436176540762905815824428826, 8.850860753316669630442735208663, 9.900357535058053880981760795741, 10.49577080681960271361559491866

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