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Properties

Label 2-756-63.20-c3-0-19
Degree $2$
Conductor $756$
Sign $-0.240 + 0.970i$
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  + (2.34 + 4.05i)5-s + (18.4 + 1.99i)7-s + (−16.1 − 9.30i)11-s + (−44.1 + 25.4i)13-s − 112.·17-s − 111. i·19-s + (124. − 71.8i)23-s + (51.5 − 89.2i)25-s + (−206. − 119. i)29-s + (179. − 103. i)31-s + (35.0 + 79.3i)35-s − 227.·37-s + (133. + 230. i)41-s + (−170. + 294. i)43-s + (−111. + 193. i)47-s + ⋯
L(s)  = 1  + (0.209 + 0.362i)5-s + (0.994 + 0.107i)7-s + (−0.441 − 0.255i)11-s + (−0.941 + 0.543i)13-s − 1.60·17-s − 1.34i·19-s + (1.12 − 0.651i)23-s + (0.412 − 0.713i)25-s + (−1.32 − 0.764i)29-s + (1.04 − 0.602i)31-s + (0.169 + 0.383i)35-s − 1.01·37-s + (0.507 + 0.878i)41-s + (−0.603 + 1.04i)43-s + (−0.347 + 0.601i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.240 + 0.970i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.240 + 0.970i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ -0.240 + 0.970i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.202298181\)
\(L(\frac12)\) \(\approx\) \(1.202298181\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.4 - 1.99i)T \)
good5 \( 1 + (-2.34 - 4.05i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (16.1 + 9.30i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (44.1 - 25.4i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 112.T + 4.91e3T^{2} \)
19 \( 1 + 111. iT - 6.85e3T^{2} \)
23 \( 1 + (-124. + 71.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (206. + 119. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-179. + 103. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 227.T + 5.06e4T^{2} \)
41 \( 1 + (-133. - 230. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (170. - 294. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (111. - 193. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 547. iT - 1.48e5T^{2} \)
59 \( 1 + (43.9 + 76.1i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-312. - 180. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (372. + 645. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 135. iT - 3.57e5T^{2} \)
73 \( 1 + 467. iT - 3.89e5T^{2} \)
79 \( 1 + (-192. + 332. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-597. + 1.03e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.38e3T + 7.04e5T^{2} \)
97 \( 1 + (1.07e3 + 617. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−9.621098693405520741681103302889, −8.869400642578877840615847598704, −8.010305639767684094549699751013, −7.02874412659018164738410075594, −6.32579756722788158441225629854, −4.88742169752310571731631107194, −4.55699996154118843451673362083, −2.78792528408653526056611715948, −2.04314774054006283092552709983, −0.31469098884592499643278212197, 1.35304322514361506884708178150, 2.38856466515036518502506543613, 3.82135494623242486991780698223, 5.05044318211685868414287318846, 5.37311767483521886453858674904, 6.91162432933512679300113661895, 7.57602994650279610510886316960, 8.553290534633356433792993851975, 9.217137968359692255779455594882, 10.34560112289057326705132852296

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