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Properties

Label 2-756-28.27-c1-0-52
Degree $2$
Conductor $756$
Sign $-0.987 - 0.157i$
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  + (−0.840 − 1.13i)2-s + (−0.588 + 1.91i)4-s − 2.67i·5-s + (−0.370 − 2.61i)7-s + (2.66 − 0.935i)8-s + (−3.04 + 2.25i)10-s − 1.39i·11-s + 3.08i·13-s + (−2.66 + 2.62i)14-s + (−3.30 − 2.25i)16-s − 5.83i·17-s − 1.91·19-s + (5.12 + 1.57i)20-s + (−1.58 + 1.17i)22-s + 1.39i·23-s + ⋯
L(s)  = 1  + (−0.593 − 0.804i)2-s + (−0.294 + 0.955i)4-s − 1.19i·5-s + (−0.140 − 0.990i)7-s + (0.943 − 0.330i)8-s + (−0.963 + 0.711i)10-s − 0.421i·11-s + 0.855i·13-s + (−0.713 + 0.700i)14-s + (−0.826 − 0.562i)16-s − 1.41i·17-s − 0.440·19-s + (1.14 + 0.352i)20-s + (−0.338 + 0.250i)22-s + 0.291i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.987 - 0.157i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.987 - 0.157i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0588481 + 0.742142i\)
\(L(\frac12)\) \(\approx\) \(0.0588481 + 0.742142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.840 + 1.13i)T \)
3 \( 1 \)
7 \( 1 + (0.370 + 2.61i)T \)
good5 \( 1 + 2.67iT - 5T^{2} \)
11 \( 1 + 1.39iT - 11T^{2} \)
13 \( 1 - 3.08iT - 13T^{2} \)
17 \( 1 + 5.83iT - 17T^{2} \)
19 \( 1 + 1.91T + 19T^{2} \)
23 \( 1 - 1.39iT - 23T^{2} \)
29 \( 1 - 4.90T + 29T^{2} \)
31 \( 1 + 9.53T + 31T^{2} \)
37 \( 1 - 6.83T + 37T^{2} \)
41 \( 1 + 0.693iT - 41T^{2} \)
43 \( 1 + 0.678iT - 43T^{2} \)
47 \( 1 + 8.26T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 12.8iT - 61T^{2} \)
67 \( 1 - 3.08iT - 67T^{2} \)
71 \( 1 + 9.79iT - 71T^{2} \)
73 \( 1 - 5.91iT - 73T^{2} \)
79 \( 1 - 12.0iT - 79T^{2} \)
83 \( 1 - 5.77T + 83T^{2} \)
89 \( 1 - 6.65iT - 89T^{2} \)
97 \( 1 - 12.8iT - 97T^{2} \)
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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

−9.599430399298720018661517536766, −9.341082870345613165462131409867, −8.355156463315594458291658197582, −7.56398438741783413976164868087, −6.60539187093932777233739710129, −4.97823869372909847837275784433, −4.32187348716115475291637445696, −3.21042629555927507767841537968, −1.64841216507231503966268518782, −0.46610427332496254512476535713, 1.94516719511839978220628874177, 3.18712306656382562087554105331, 4.69160491602755826780872974618, 5.95038363596575300174312347059, 6.31015998325641479019130514327, 7.37708095230519101954143523118, 8.155937696954146076006765358561, 8.972969334771534141271043013705, 9.945459500206929190908131003803, 10.59033115731550216062599443273

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