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Properties

Label 2-756-21.17-c3-0-16
Degree $2$
Conductor $756$
Sign $0.568 + 0.823i$
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  + (−2.90 − 5.03i)5-s + (−16.4 + 8.50i)7-s + (30.5 + 17.6i)11-s − 5.07i·13-s + (39.0 − 67.5i)17-s + (−47.6 + 27.4i)19-s + (−164. + 95.0i)23-s + (45.5 − 78.9i)25-s + 189. i·29-s + (71.5 + 41.3i)31-s + (90.6 + 58.0i)35-s + (−116. − 201. i)37-s + 489.·41-s + 361.·43-s + (−262. − 455. i)47-s + ⋯
L(s)  = 1  + (−0.260 − 0.450i)5-s + (−0.888 + 0.459i)7-s + (0.836 + 0.482i)11-s − 0.108i·13-s + (0.556 − 0.964i)17-s + (−0.574 + 0.331i)19-s + (−1.49 + 0.861i)23-s + (0.364 − 0.631i)25-s + 1.21i·29-s + (0.414 + 0.239i)31-s + (0.437 + 0.280i)35-s + (−0.516 − 0.894i)37-s + 1.86·41-s + 1.28·43-s + (−0.815 − 1.41i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.823i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.568 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.568 + 0.823i$
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: \((2,\ 756,\ (\ :3/2),\ 0.568 + 0.823i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.421782016\)
\(L(\frac12)\) \(\approx\) \(1.421782016\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.4 - 8.50i)T \)
good5 \( 1 + (2.90 + 5.03i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-30.5 - 17.6i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 5.07iT - 2.19e3T^{2} \)
17 \( 1 + (-39.0 + 67.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (47.6 - 27.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (164. - 95.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 189. iT - 2.43e4T^{2} \)
31 \( 1 + (-71.5 - 41.3i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (116. + 201. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 489.T + 6.89e4T^{2} \)
43 \( 1 - 361.T + 7.95e4T^{2} \)
47 \( 1 + (262. + 455. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (278. + 160. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (172. - 298. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-673. + 388. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-198. + 343. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 871. iT - 3.57e5T^{2} \)
73 \( 1 + (32.1 + 18.5i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (106. + 184. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 376.T + 5.71e5T^{2} \)
89 \( 1 + (-770. - 1.33e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.44e3iT - 9.12e5T^{2} \)
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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

−9.632054353362111702065223319265, −9.134657120072695008402195040349, −8.152216458681380082281961445909, −7.17888852841673300041380463468, −6.29019796036914771750049265112, −5.39069834696300723645155053219, −4.24538164416546215986748668004, −3.32108497516856722372099104147, −1.99749914299771497027421996198, −0.49891730805791051908106603367, 0.914218190151168314505856247951, 2.53081671720290437958674404077, 3.68956384457950748123000646821, 4.31241825233387518068312306069, 6.08283870154911111032590679545, 6.34758262252166702010263958627, 7.48630565581433926310802038114, 8.318721444941694115906575731323, 9.335204416717074474466672440558, 10.09986132822239997150239512693

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