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Properties

Label 2-756-63.20-c3-0-4
Degree $2$
Conductor $756$
Sign $-0.0966 - 0.995i$
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  + (−2.20 − 3.82i)5-s + (−17.7 − 5.43i)7-s + (59.1 + 34.1i)11-s + (−29.9 + 17.3i)13-s + 21.8·17-s − 124. i·19-s + (−61.3 + 35.4i)23-s + (52.7 − 91.3i)25-s + (−187. − 108. i)29-s + (−242. + 140. i)31-s + (18.2 + 79.6i)35-s + 150.·37-s + (136. + 236. i)41-s + (−136. + 236. i)43-s + (97.3 − 168. i)47-s + ⋯
L(s)  = 1  + (−0.197 − 0.341i)5-s + (−0.955 − 0.293i)7-s + (1.62 + 0.936i)11-s + (−0.639 + 0.369i)13-s + 0.311·17-s − 1.50i·19-s + (−0.556 + 0.321i)23-s + (0.422 − 0.731i)25-s + (−1.20 − 0.694i)29-s + (−1.40 + 0.811i)31-s + (0.0882 + 0.384i)35-s + 0.670·37-s + (0.519 + 0.900i)41-s + (−0.484 + 0.839i)43-s + (0.302 − 0.523i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0966 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0966 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.0966 - 0.995i$
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: \((2,\ 756,\ (\ :3/2),\ -0.0966 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9910123436\)
\(L(\frac12)\) \(\approx\) \(0.9910123436\)
\(L(\frac{5}{2})\) 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 \( 1 + (17.7 + 5.43i)T \)
good5 \( 1 + (2.20 + 3.82i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-59.1 - 34.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (29.9 - 17.3i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 21.8T + 4.91e3T^{2} \)
19 \( 1 + 124. iT - 6.85e3T^{2} \)
23 \( 1 + (61.3 - 35.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (187. + 108. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (242. - 140. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 150.T + 5.06e4T^{2} \)
41 \( 1 + (-136. - 236. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (136. - 236. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-97.3 + 168. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 520. iT - 1.48e5T^{2} \)
59 \( 1 + (-301. - 521. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (145. + 84.0i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-371. - 643. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 758. iT - 3.57e5T^{2} \)
73 \( 1 - 1.15e3iT - 3.89e5T^{2} \)
79 \( 1 + (-78.6 + 136. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (137. - 237. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.04e3T + 7.04e5T^{2} \)
97 \( 1 + (211. + 122. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−9.802165345188183243135477167804, −9.506282375155567374163906020039, −8.673168354310619092782296814470, −7.25677634229983148037508764539, −6.91305984963835984913211230287, −5.83379667737742820584320691636, −4.51311448372315961150342426824, −3.89996617859019065188353272080, −2.54127304045920467436141358470, −1.12683012655579278508924989669, 0.29738566941763425081754144018, 1.86279809283250806195090337203, 3.43837049217165082382575700084, 3.75570770575701494029876643930, 5.49337523098374850864634746136, 6.16042338166377455051806807944, 7.04538885362191788009714939705, 7.963999045712656613460782485897, 9.100027527422002542607104895613, 9.549261802479453574447503041263

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