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Properties

Label 2-756-63.41-c3-0-6
Degree $2$
Conductor $756$
Sign $-0.247 - 0.969i$
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  + (−2.99 + 5.19i)5-s + (−0.375 + 18.5i)7-s + (39.3 − 22.7i)11-s + (−22.7 − 13.1i)13-s + 19.7·17-s + 27.9i·19-s + (60.3 + 34.8i)23-s + (44.5 + 77.0i)25-s + (119. − 68.7i)29-s + (138. + 79.7i)31-s + (−95.0 − 57.4i)35-s − 287.·37-s + (20.4 − 35.3i)41-s + (55.4 + 95.9i)43-s + (−109. − 189. i)47-s + ⋯
L(s)  = 1  + (−0.268 + 0.464i)5-s + (−0.0202 + 0.999i)7-s + (1.07 − 0.623i)11-s + (−0.485 − 0.280i)13-s + 0.281·17-s + 0.337i·19-s + (0.547 + 0.315i)23-s + (0.356 + 0.616i)25-s + (0.762 − 0.440i)29-s + (0.800 + 0.462i)31-s + (−0.459 − 0.277i)35-s − 1.27·37-s + (0.0778 − 0.134i)41-s + (0.196 + 0.340i)43-s + (−0.340 − 0.589i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.247 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.247 - 0.969i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ -0.247 - 0.969i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.634896051\)
\(L(\frac12)\) \(\approx\) \(1.634896051\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.375 - 18.5i)T \)
good5 \( 1 + (2.99 - 5.19i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-39.3 + 22.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (22.7 + 13.1i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 19.7T + 4.91e3T^{2} \)
19 \( 1 - 27.9iT - 6.85e3T^{2} \)
23 \( 1 + (-60.3 - 34.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-119. + 68.7i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-138. - 79.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 287.T + 5.06e4T^{2} \)
41 \( 1 + (-20.4 + 35.3i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-55.4 - 95.9i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (109. + 189. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 209. iT - 1.48e5T^{2} \)
59 \( 1 + (413. - 716. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (594. - 343. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (171. - 296. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 387. iT - 3.57e5T^{2} \)
73 \( 1 + 220. iT - 3.89e5T^{2} \)
79 \( 1 + (-242. - 419. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-354. - 613. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 140.T + 7.04e5T^{2} \)
97 \( 1 + (1.30e3 - 753. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−10.21705075257007703692556001533, −9.206999798983143649296089557361, −8.625901712113704589869032611934, −7.60494893789493060442663420886, −6.64086831078480153266951411919, −5.84754431856934146331149176492, −4.85032478824373498366141554108, −3.53700569483824358794874194798, −2.72679709293993760298002977419, −1.25553673798383395478376855983, 0.48973607896895457011897803504, 1.62343863522668128806883589209, 3.20847822584943787671087025291, 4.36914937479045074512386291335, 4.84706342560675164313739187157, 6.43162040194403195507064452864, 7.01968476410284290862581722843, 7.956185313326976557675874790938, 8.898012740269084243840230431685, 9.706909680962453338824707755372

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