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Properties

Label 2-756-12.11-c1-0-0
Degree $2$
Conductor $756$
Sign $-0.585 + 0.810i$
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  + (−1.25 + 0.644i)2-s + (1.17 − 1.62i)4-s + 3.70i·5-s − i·7-s + (−0.429 + 2.79i)8-s + (−2.38 − 4.66i)10-s − 3.09·11-s − 5.37·13-s + (0.644 + 1.25i)14-s + (−1.26 − 3.79i)16-s − 6.77i·17-s + 3.99i·19-s + (6.01 + 4.34i)20-s + (3.90 − 1.99i)22-s − 2.22·23-s + ⋯
L(s)  = 1  + (−0.890 + 0.455i)2-s + (0.585 − 0.810i)4-s + 1.65i·5-s − 0.377i·7-s + (−0.151 + 0.988i)8-s + (−0.755 − 1.47i)10-s − 0.934·11-s − 1.48·13-s + (0.172 + 0.336i)14-s + (−0.315 − 0.949i)16-s − 1.64i·17-s + 0.917i·19-s + (1.34 + 0.970i)20-s + (0.831 − 0.425i)22-s − 0.464·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.585 + 0.810i$
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: \((2,\ 756,\ (\ :1/2),\ -0.585 + 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0208543 - 0.0407669i\)
\(L(\frac12)\) \(\approx\) \(0.0208543 - 0.0407669i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 - 0.644i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 3.70iT - 5T^{2} \)
11 \( 1 + 3.09T + 11T^{2} \)
13 \( 1 + 5.37T + 13T^{2} \)
17 \( 1 + 6.77iT - 17T^{2} \)
19 \( 1 - 3.99iT - 19T^{2} \)
23 \( 1 + 2.22T + 23T^{2} \)
29 \( 1 - 1.70iT - 29T^{2} \)
31 \( 1 + 5.12iT - 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 - 0.313iT - 41T^{2} \)
43 \( 1 + 3.71iT - 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 - 2.32iT - 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 + 5.02T + 61T^{2} \)
67 \( 1 + 10.0iT - 67T^{2} \)
71 \( 1 + 1.38T + 71T^{2} \)
73 \( 1 + 1.59T + 73T^{2} \)
79 \( 1 - 13.0iT - 79T^{2} \)
83 \( 1 - 5.76T + 83T^{2} \)
89 \( 1 + 7.23iT - 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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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

−10.71703770744912965208940439758, −9.851491599493625852477106838478, −9.609897542792070764655100152041, −7.88033850752494990233090968605, −7.56408132308658656562410556348, −6.82414511547226913512722744806, −5.90474813508470669779645568783, −4.79497252551556498186855117657, −3.02000607217532291881152803936, −2.28671549986514988770822228440, 0.02917744647176122734146497480, 1.59532293597945130369101068709, 2.73847228658102649943016791634, 4.32139147896337872357123697049, 5.12816534211576086854228172160, 6.30097236397894056781717510771, 7.70688972228809996460198101829, 8.164485399604847742407306392507, 8.996764411536421565101764405640, 9.675535383655811424109917416712

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