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Properties

Label 2-176-44.43-c5-0-22
Degree $2$
Conductor $176$
Sign $0.959 + 0.280i$
Analytic cond. $28.2275$
Root an. cond. $5.31296$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.663i·3-s + 68.6·5-s + 214.·7-s + 242.·9-s + (−95.0 − 389. i)11-s − 556. i·13-s + 45.5i·15-s + 1.84e3i·17-s − 1.21e3·19-s + 142. i·21-s − 121. i·23-s + 1.59e3·25-s + 322. i·27-s − 4.26e3i·29-s − 3.81e3i·31-s + ⋯
L(s)  = 1  + 0.0425i·3-s + 1.22·5-s + 1.65·7-s + 0.998·9-s + (−0.236 − 0.971i)11-s − 0.912i·13-s + 0.0523i·15-s + 1.55i·17-s − 0.772·19-s + 0.0704i·21-s − 0.0478i·23-s + 0.509·25-s + 0.0850i·27-s − 0.942i·29-s − 0.712i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.959 + 0.280i$
Analytic conductor: \(28.2275\)
Root analytic conductor: \(5.31296\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :5/2),\ 0.959 + 0.280i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.146604468\)
\(L(\frac12)\) \(\approx\) \(3.146604468\)
\(L(\frac{7}{2})\) 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 \( 1 + (95.0 + 389. i)T \)
good3 \( 1 - 0.663iT - 243T^{2} \)
5 \( 1 - 68.6T + 3.12e3T^{2} \)
7 \( 1 - 214.T + 1.68e4T^{2} \)
13 \( 1 + 556. iT - 3.71e5T^{2} \)
17 \( 1 - 1.84e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.21e3T + 2.47e6T^{2} \)
23 \( 1 + 121. iT - 6.43e6T^{2} \)
29 \( 1 + 4.26e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.81e3iT - 2.86e7T^{2} \)
37 \( 1 - 9.38e3T + 6.93e7T^{2} \)
41 \( 1 - 1.48e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.82e4T + 1.47e8T^{2} \)
47 \( 1 + 2.42e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.80e4T + 4.18e8T^{2} \)
59 \( 1 - 970. iT - 7.14e8T^{2} \)
61 \( 1 - 4.83e4iT - 8.44e8T^{2} \)
67 \( 1 + 9.34e3iT - 1.35e9T^{2} \)
71 \( 1 - 4.24e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.18e3iT - 2.07e9T^{2} \)
79 \( 1 + 4.97e4T + 3.07e9T^{2} \)
83 \( 1 - 4.08e4T + 3.93e9T^{2} \)
89 \( 1 + 4.75e3T + 5.58e9T^{2} \)
97 \( 1 - 4.44e4T + 8.58e9T^{2} \)
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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

−11.62607935489166799094571656512, −10.59512636448497832502263847967, −10.02543727399954538091470715772, −8.532139669415176399669783672694, −7.88476031032302004235905838197, −6.25093504625275879612135336707, −5.38489657388937217309436291344, −4.14759072734806914158611171812, −2.20582863164656538486421916617, −1.20128831867993346085127433242, 1.47102498887444616750185326078, 2.17829154922091279569060625761, 4.51432447796725327916328229154, 5.13912228732982786792003259594, 6.71918335104891931419054879438, 7.59604171603703176970584659974, 8.981763396671098018425063798523, 9.831111191739539041898371420527, 10.79029142383358395732323646309, 11.84417028925780767111457830514

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