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Properties

Label 2-756-21.17-c3-0-11
Degree $2$
Conductor $756$
Sign $0.320 - 0.947i$
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  + (5.19 + 9.00i)5-s + (−16.6 − 8.10i)7-s + (53.0 + 30.6i)11-s − 12.9i·13-s + (17.2 − 29.8i)17-s + (−29.2 + 16.8i)19-s + (37.4 − 21.6i)23-s + (8.44 − 14.6i)25-s + 62.7i·29-s + (2.36 + 1.36i)31-s + (−13.5 − 192. i)35-s + (162. + 281. i)37-s − 274.·41-s + 42.0·43-s + (103. + 179. i)47-s + ⋯
L(s)  = 1  + (0.465 + 0.805i)5-s + (−0.899 − 0.437i)7-s + (1.45 + 0.839i)11-s − 0.277i·13-s + (0.246 − 0.426i)17-s + (−0.352 + 0.203i)19-s + (0.339 − 0.196i)23-s + (0.0675 − 0.116i)25-s + 0.402i·29-s + (0.0136 + 0.00790i)31-s + (−0.0656 − 0.927i)35-s + (0.723 + 1.25i)37-s − 1.04·41-s + 0.149·43-s + (0.321 + 0.557i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.320 - 0.947i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ 0.320 - 0.947i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.979311798\)
\(L(\frac12)\) \(\approx\) \(1.979311798\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.6 + 8.10i)T \)
good5 \( 1 + (-5.19 - 9.00i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-53.0 - 30.6i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 12.9iT - 2.19e3T^{2} \)
17 \( 1 + (-17.2 + 29.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (29.2 - 16.8i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-37.4 + 21.6i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 62.7iT - 2.43e4T^{2} \)
31 \( 1 + (-2.36 - 1.36i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-162. - 281. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 274.T + 6.89e4T^{2} \)
43 \( 1 - 42.0T + 7.95e4T^{2} \)
47 \( 1 + (-103. - 179. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-174. - 100. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-29.4 + 50.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-176. + 101. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (288. - 499. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 658. iT - 3.57e5T^{2} \)
73 \( 1 + (-414. - 239. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-109. - 189. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.03e3T + 5.71e5T^{2} \)
89 \( 1 + (-347. - 601. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 498. iT - 9.12e5T^{2} \)
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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

−9.971302200736841412975698932912, −9.562477131378916371378383973042, −8.501553708389372682725532391055, −7.14604688793503941587453016384, −6.74483589203830851479954540122, −5.94816301633025762718855473134, −4.55449009695206524431316121088, −3.55244166961102238194581184499, −2.55882731826056423465880249974, −1.13796344091312212568622214545, 0.61363081667639528766752435429, 1.82287429097849120517522064615, 3.26944264231443439020510878864, 4.19932875060110366972459226690, 5.48498954513344070575869255796, 6.16899685158976924631155697587, 6.96867489053227994263163727315, 8.352193183177152338525843369853, 9.120255541079288572772560526577, 9.434229317768634894508448525584

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