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Properties

Label 2-756-12.11-c1-0-9
Degree $2$
Conductor $756$
Sign $0.903 - 0.428i$
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  + (−0.310 − 1.37i)2-s + (−1.80 + 0.856i)4-s + 1.05i·5-s − i·7-s + (1.74 + 2.22i)8-s + (1.45 − 0.327i)10-s − 2.97·11-s + 0.985·13-s + (−1.37 + 0.310i)14-s + (2.53 − 3.09i)16-s + 6.35i·17-s + 1.26i·19-s + (−0.902 − 1.90i)20-s + (0.924 + 4.11i)22-s − 2.66·23-s + ⋯
L(s)  = 1  + (−0.219 − 0.975i)2-s + (−0.903 + 0.428i)4-s + 0.471i·5-s − 0.377i·7-s + (0.616 + 0.787i)8-s + (0.459 − 0.103i)10-s − 0.898·11-s + 0.273·13-s + (−0.368 + 0.0829i)14-s + (0.633 − 0.773i)16-s + 1.54i·17-s + 0.291i·19-s + (−0.201 − 0.425i)20-s + (0.197 + 0.876i)22-s − 0.555·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.903 - 0.428i$
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: \((2,\ 756,\ (\ :1/2),\ 0.903 - 0.428i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.929165 + 0.209031i\)
\(L(\frac12)\) \(\approx\) \(0.929165 + 0.209031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.310 + 1.37i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 1.05iT - 5T^{2} \)
11 \( 1 + 2.97T + 11T^{2} \)
13 \( 1 - 0.985T + 13T^{2} \)
17 \( 1 - 6.35iT - 17T^{2} \)
19 \( 1 - 1.26iT - 19T^{2} \)
23 \( 1 + 2.66T + 23T^{2} \)
29 \( 1 - 5.12iT - 29T^{2} \)
31 \( 1 - 8.41iT - 31T^{2} \)
37 \( 1 - 8.91T + 37T^{2} \)
41 \( 1 - 4.54iT - 41T^{2} \)
43 \( 1 - 0.646iT - 43T^{2} \)
47 \( 1 + 4.64T + 47T^{2} \)
53 \( 1 - 2.57iT - 53T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 - 4.89T + 61T^{2} \)
67 \( 1 - 8.21iT - 67T^{2} \)
71 \( 1 + 0.583T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 + 15.1iT - 79T^{2} \)
83 \( 1 + 4.73T + 83T^{2} \)
89 \( 1 + 1.74iT - 89T^{2} \)
97 \( 1 - 3.00T + 97T^{2} \)
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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

−10.49171455878446270175801171517, −9.910679629794377350908428362866, −8.666702446956855534027583549805, −8.113101175644363143231111963127, −7.07906336979209603422575720343, −5.87817173798825031523850257217, −4.73686442209325509834752448582, −3.69705421969304574226016961848, −2.77288426290289091874076829384, −1.42790529679813966379091198282, 0.56318816293285337710607839639, 2.54383516523869391698099217417, 4.17473564421522627114109209801, 5.10537996359516597814274315433, 5.79215636881736051879923697592, 6.85504436728025141493601965491, 7.78220370591953005320459247183, 8.408543881482754227962228565640, 9.386604840658641233440694043111, 9.880495392980252202472959031424

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