Properties

Label 2-756-12.11-c1-0-45
Degree $2$
Conductor $756$
Sign $-0.668 - 0.743i$
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  + (0.575 − 1.29i)2-s + (−1.33 − 1.48i)4-s − 0.311i·5-s + i·7-s + (−2.69 + 0.870i)8-s + (−0.402 − 0.179i)10-s − 3.59·11-s − 6.44·13-s + (1.29 + 0.575i)14-s + (−0.425 + 3.97i)16-s − 2.96i·17-s + 0.684i·19-s + (−0.463 + 0.416i)20-s + (−2.06 + 4.64i)22-s − 7.58·23-s + ⋯
L(s)  = 1  + (0.407 − 0.913i)2-s + (−0.668 − 0.743i)4-s − 0.139i·5-s + 0.377i·7-s + (−0.951 + 0.307i)8-s + (−0.127 − 0.0566i)10-s − 1.08·11-s − 1.78·13-s + (0.345 + 0.153i)14-s + (−0.106 + 0.994i)16-s − 0.718i·17-s + 0.157i·19-s + (−0.103 + 0.0930i)20-s + (−0.441 + 0.989i)22-s − 1.58·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.668 - 0.743i$
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: \((2,\ 756,\ (\ :1/2),\ -0.668 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147795 + 0.331521i\)
\(L(\frac12)\) \(\approx\) \(0.147795 + 0.331521i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.575 + 1.29i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 + 0.311iT - 5T^{2} \)
11 \( 1 + 3.59T + 11T^{2} \)
13 \( 1 + 6.44T + 13T^{2} \)
17 \( 1 + 2.96iT - 17T^{2} \)
19 \( 1 - 0.684iT - 19T^{2} \)
23 \( 1 + 7.58T + 23T^{2} \)
29 \( 1 + 9.26iT - 29T^{2} \)
31 \( 1 - 1.91iT - 31T^{2} \)
37 \( 1 + 8.40T + 37T^{2} \)
41 \( 1 + 7.50iT - 41T^{2} \)
43 \( 1 - 9.30iT - 43T^{2} \)
47 \( 1 - 9.82T + 47T^{2} \)
53 \( 1 - 5.43iT - 53T^{2} \)
59 \( 1 + 2.29T + 59T^{2} \)
61 \( 1 - 0.486T + 61T^{2} \)
67 \( 1 - 1.09iT - 67T^{2} \)
71 \( 1 + 1.76T + 71T^{2} \)
73 \( 1 - 3.32T + 73T^{2} \)
79 \( 1 - 1.39iT - 79T^{2} \)
83 \( 1 + 0.283T + 83T^{2} \)
89 \( 1 + 8.91iT - 89T^{2} \)
97 \( 1 - 0.621T + 97T^{2} \)
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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

−9.993155595588284315144244924764, −9.248130549330476135549553984136, −8.196435860505689911710730081412, −7.26622923075764695394367921919, −5.88178236568537734686401419224, −5.11059357351348712918111748469, −4.32403072556945479286109919970, −2.86161514208640678169017413969, −2.17118335378283223458549589691, −0.14786381382789634285743781993, 2.45932761999878677505500660668, 3.66697157783992748468611354385, 4.83373783257520615156168069589, 5.42646156185417258657024243013, 6.64789757452093249766754306069, 7.36986771113401326034380472943, 8.047217294428024488829422378437, 8.991414824758269037461114430680, 10.06231093450915705845537227956, 10.63632024708353233238137087559

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