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Properties

Label 2-176-11.4-c3-0-10
Degree $2$
Conductor $176$
Sign $0.124 + 0.992i$
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  + (−2.64 + 8.12i)3-s + (−10.3 + 7.48i)5-s + (−7.24 − 22.3i)7-s + (−37.2 − 27.0i)9-s + (3.69 + 36.2i)11-s + (−9.27 − 6.73i)13-s + (−33.6 − 103. i)15-s + (52.9 − 38.5i)17-s + (2.24 − 6.89i)19-s + 200.·21-s + 104.·23-s + (11.4 − 35.3i)25-s + (131. − 95.5i)27-s + (−39.3 − 121. i)29-s + (−233. − 169. i)31-s + ⋯
L(s)  = 1  + (−0.508 + 1.56i)3-s + (−0.921 + 0.669i)5-s + (−0.391 − 1.20i)7-s + (−1.37 − 1.00i)9-s + (0.101 + 0.994i)11-s + (−0.197 − 0.143i)13-s + (−0.578 − 1.78i)15-s + (0.756 − 0.549i)17-s + (0.0270 − 0.0832i)19-s + 2.08·21-s + 0.943·23-s + (0.0919 − 0.283i)25-s + (0.937 − 0.680i)27-s + (−0.251 − 0.775i)29-s + (−1.35 − 0.983i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.124 + 0.992i$
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: \((2,\ 176,\ (\ :3/2),\ 0.124 + 0.992i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.100564 - 0.0886943i\)
\(L(\frac12)\) \(\approx\) \(0.100564 - 0.0886943i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.69 - 36.2i)T \)
good3 \( 1 + (2.64 - 8.12i)T + (-21.8 - 15.8i)T^{2} \)
5 \( 1 + (10.3 - 7.48i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (7.24 + 22.3i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (9.27 + 6.73i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (-52.9 + 38.5i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (-2.24 + 6.89i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 - 104.T + 1.21e4T^{2} \)
29 \( 1 + (39.3 + 121. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (233. + 169. i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (26.3 + 81.2i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (41.8 - 128. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 + 353.T + 7.95e4T^{2} \)
47 \( 1 + (-41.5 + 128. i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (405. + 294. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (201. + 619. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (295. - 215. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 - 294.T + 3.00e5T^{2} \)
71 \( 1 + (-107. + 77.8i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (-145. - 446. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (-330. - 239. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (1.09e3 - 799. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 + 260.T + 7.04e5T^{2} \)
97 \( 1 + (1.14e3 + 831. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−11.56143704230969798918394235612, −10.98715757247246937251182872651, −10.03908589066704894538249570217, −9.495731873035931746632353032957, −7.71213034566876046368668357639, −6.85248875248973092086266739929, −5.18029375958443141764921577318, −4.11397546174268328428678810638, −3.36750562196320944481264282089, −0.06841925235016306106376670684, 1.41205310385267054220955367462, 3.18541639119534585332478531343, 5.25190155933358114982794977689, 6.14328044050704840046345250563, 7.29081698544978753413827140922, 8.309947393404431134563729020312, 9.003806277432711764155430469731, 10.93367091061604485244615930876, 11.89123782185050386896482060578, 12.39963569962743292646025402642

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