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Properties

Label 2-176-11.4-c3-0-7
Degree $2$
Conductor $176$
Sign $0.702 - 0.711i$
Analytic cond. $10.3843$
Root an. cond. $3.22247$
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  + (−1.40 + 4.31i)3-s + (6.98 − 5.07i)5-s + (−0.513 − 1.58i)7-s + (5.15 + 3.74i)9-s + (26.2 − 25.3i)11-s + (23.1 + 16.8i)13-s + (12.1 + 37.2i)15-s + (−6.36 + 4.62i)17-s + (−14.8 + 45.6i)19-s + 7.55·21-s + 153.·23-s + (−15.5 + 47.9i)25-s + (−122. + 89.0i)27-s + (74.9 + 230. i)29-s + (134. + 97.5i)31-s + ⋯
L(s)  = 1  + (−0.270 + 0.831i)3-s + (0.624 − 0.453i)5-s + (−0.0277 − 0.0853i)7-s + (0.191 + 0.138i)9-s + (0.719 − 0.694i)11-s + (0.494 + 0.358i)13-s + (0.208 + 0.642i)15-s + (−0.0907 + 0.0659i)17-s + (−0.179 + 0.551i)19-s + 0.0784·21-s + 1.38·23-s + (−0.124 + 0.383i)25-s + (−0.874 + 0.635i)27-s + (0.479 + 1.47i)29-s + (0.777 + 0.565i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.702 - 0.711i$
Analytic conductor: \(10.3843\)
Root analytic conductor: \(3.22247\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :3/2),\ 0.702 - 0.711i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.72899 + 0.722719i\)
\(L(\frac12)\) \(\approx\) \(1.72899 + 0.722719i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-26.2 + 25.3i)T \)
good3 \( 1 + (1.40 - 4.31i)T + (-21.8 - 15.8i)T^{2} \)
5 \( 1 + (-6.98 + 5.07i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (0.513 + 1.58i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (-23.1 - 16.8i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (6.36 - 4.62i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (14.8 - 45.6i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 - 153.T + 1.21e4T^{2} \)
29 \( 1 + (-74.9 - 230. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (-134. - 97.5i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (21.2 + 65.4i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-95.0 + 292. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 - 52.9T + 7.95e4T^{2} \)
47 \( 1 + (-16.8 + 51.8i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (202. + 147. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (117. + 360. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (565. - 410. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 - 278.T + 3.00e5T^{2} \)
71 \( 1 + (125. - 90.9i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (293. + 904. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (-1.79 - 1.30i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (614. - 446. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 - 1.15e3T + 7.04e5T^{2} \)
97 \( 1 + (598. + 434. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−12.37092929115705988400771062029, −11.12749435649398104261098946734, −10.44946805725019672686556672318, −9.312693352029220144569377734363, −8.676698134155626009786665060885, −7.00956458939355372400942678100, −5.77300409469860833434716909943, −4.75686623831475444488671313770, −3.51586304633171918568126536394, −1.39541315420421223589546316749, 1.09704363425571997014807827331, 2.57805284299659068619503787198, 4.41267254776322730248806693788, 6.09233771870732441037162049427, 6.68767139721973030759874252437, 7.77884738393368934365487463425, 9.192360627255633597184395922227, 10.09003529336872532632770617681, 11.27387381492029149146088085126, 12.18779484134854961510098585901

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