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Properties

Label 2-756-21.20-c3-0-0
Degree $2$
Conductor $756$
Sign $-0.985 + 0.171i$
Analytic cond. $44.6054$
Root an. cond. $6.67873$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.67·5-s + (−18.2 + 3.18i)7-s + 34.9i·11-s + 63.3i·13-s + 32.0·17-s − 125. i·19-s + 63.6i·23-s − 80.5·25-s − 303. i·29-s + 5.46i·31-s + (−121. + 21.2i)35-s + 45.0·37-s − 185.·41-s − 224.·43-s − 534.·47-s + ⋯
L(s)  = 1  + 0.596·5-s + (−0.985 + 0.171i)7-s + 0.957i·11-s + 1.35i·13-s + 0.457·17-s − 1.51i·19-s + 0.577i·23-s − 0.644·25-s − 1.94i·29-s + 0.0316i·31-s + (−0.587 + 0.102i)35-s + 0.200·37-s − 0.708·41-s − 0.797·43-s − 1.65·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.171i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.985 + 0.171i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ -0.985 + 0.171i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1684384048\)
\(L(\frac12)\) \(\approx\) \(0.1684384048\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (18.2 - 3.18i)T \)
good5 \( 1 - 6.67T + 125T^{2} \)
11 \( 1 - 34.9iT - 1.33e3T^{2} \)
13 \( 1 - 63.3iT - 2.19e3T^{2} \)
17 \( 1 - 32.0T + 4.91e3T^{2} \)
19 \( 1 + 125. iT - 6.85e3T^{2} \)
23 \( 1 - 63.6iT - 1.21e4T^{2} \)
29 \( 1 + 303. iT - 2.43e4T^{2} \)
31 \( 1 - 5.46iT - 2.97e4T^{2} \)
37 \( 1 - 45.0T + 5.06e4T^{2} \)
41 \( 1 + 185.T + 6.89e4T^{2} \)
43 \( 1 + 224.T + 7.95e4T^{2} \)
47 \( 1 + 534.T + 1.03e5T^{2} \)
53 \( 1 - 373. iT - 1.48e5T^{2} \)
59 \( 1 + 701.T + 2.05e5T^{2} \)
61 \( 1 + 74.5iT - 2.26e5T^{2} \)
67 \( 1 - 425.T + 3.00e5T^{2} \)
71 \( 1 + 936. iT - 3.57e5T^{2} \)
73 \( 1 - 957. iT - 3.89e5T^{2} \)
79 \( 1 + 129.T + 4.93e5T^{2} \)
83 \( 1 + 719.T + 5.71e5T^{2} \)
89 \( 1 + 274.T + 7.04e5T^{2} \)
97 \( 1 + 1.35e3iT - 9.12e5T^{2} \)
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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

−10.01884347642346180593075064260, −9.636752220054975150547499822317, −9.001966424552448220890959044190, −7.70244953876922751859477470289, −6.75884941433796954666894851114, −6.18915284482207245552738554032, −5.00342927958026826520400169694, −4.02940404958813724268136529637, −2.72438451795062612230855422431, −1.73167176159753396182963682256, 0.04374294762299419723920826598, 1.42357910474186637251231126710, 3.03230912447682203751448514994, 3.59154755976488573408930891563, 5.25223015220326991876975426272, 5.91559452350883421402640272988, 6.69709803396988334517817634652, 7.902822803217165502870053551926, 8.586357608678178996016541240153, 9.741099552014660458792394401647

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