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Properties

Label 2-1152-48.35-c3-0-24
Degree $2$
Conductor $1152$
Sign $0.746 - 0.664i$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
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  + (9.40 + 9.40i)5-s − 3.57·7-s + (3.36 − 3.36i)11-s + (−26.9 − 26.9i)13-s − 12.7i·17-s + (50.0 − 50.0i)19-s + 208. i·23-s + 51.7i·25-s + (134. − 134. i)29-s + 80.1i·31-s + (−33.5 − 33.5i)35-s + (308. − 308. i)37-s + 172.·41-s + (87.0 + 87.0i)43-s + 525.·47-s + ⋯
L(s)  = 1  + (0.840 + 0.840i)5-s − 0.192·7-s + (0.0921 − 0.0921i)11-s + (−0.574 − 0.574i)13-s − 0.182i·17-s + (0.604 − 0.604i)19-s + 1.88i·23-s + 0.413i·25-s + (0.859 − 0.859i)29-s + 0.464i·31-s + (−0.162 − 0.162i)35-s + (1.37 − 1.37i)37-s + 0.658·41-s + (0.308 + 0.308i)43-s + 1.63·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.664i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.746 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.746 - 0.664i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ 0.746 - 0.664i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.432059784\)
\(L(\frac12)\) \(\approx\) \(2.432059784\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5 \( 1 + (-9.40 - 9.40i)T + 125iT^{2} \)
7 \( 1 + 3.57T + 343T^{2} \)
11 \( 1 + (-3.36 + 3.36i)T - 1.33e3iT^{2} \)
13 \( 1 + (26.9 + 26.9i)T + 2.19e3iT^{2} \)
17 \( 1 + 12.7iT - 4.91e3T^{2} \)
19 \( 1 + (-50.0 + 50.0i)T - 6.85e3iT^{2} \)
23 \( 1 - 208. iT - 1.21e4T^{2} \)
29 \( 1 + (-134. + 134. i)T - 2.43e4iT^{2} \)
31 \( 1 - 80.1iT - 2.97e4T^{2} \)
37 \( 1 + (-308. + 308. i)T - 5.06e4iT^{2} \)
41 \( 1 - 172.T + 6.89e4T^{2} \)
43 \( 1 + (-87.0 - 87.0i)T + 7.95e4iT^{2} \)
47 \( 1 - 525.T + 1.03e5T^{2} \)
53 \( 1 + (-127. - 127. i)T + 1.48e5iT^{2} \)
59 \( 1 + (172. - 172. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-332. - 332. i)T + 2.26e5iT^{2} \)
67 \( 1 + (556. - 556. i)T - 3.00e5iT^{2} \)
71 \( 1 - 450. iT - 3.57e5T^{2} \)
73 \( 1 - 797. iT - 3.89e5T^{2} \)
79 \( 1 + 70.1iT - 4.93e5T^{2} \)
83 \( 1 + (636. + 636. i)T + 5.71e5iT^{2} \)
89 \( 1 + 925.T + 7.04e5T^{2} \)
97 \( 1 - 1.26e3T + 9.12e5T^{2} \)
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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

−9.706159948564180403592883217832, −8.880262329613339503292786652653, −7.59577639290876077780687842457, −7.15438081446595277847544830478, −6.00409444978813612070688863266, −5.54300097566194449264727891004, −4.27519954281422862365577759237, −3.02732648082631698189778574638, −2.39196513215901623578002626751, −0.935966323915338982312589997751, 0.72442596123906724439007240944, 1.83214417548540788361154331769, 2.88713363530739843672539366378, 4.32187476326069751203348897883, 4.96949308039925560630371197540, 5.98697976931463608823960770287, 6.66382755885341393130298810215, 7.77255542897684624933046218285, 8.653397466032121059733729657958, 9.357261189630447446483195122348

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