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Properties

Label 2-176-16.13-c1-0-8
Degree $2$
Conductor $176$
Sign $0.506 - 0.862i$
Analytic cond. $1.40536$
Root an. cond. $1.18548$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.28 + 0.586i)2-s + (0.586 + 0.586i)3-s + (1.31 + 1.50i)4-s + (−1.32 + 1.32i)5-s + (0.411 + 1.09i)6-s − 1.42i·7-s + (0.805 + 2.71i)8-s − 2.31i·9-s + (−2.47 + 0.926i)10-s + (−0.707 + 0.707i)11-s + (−0.114 + 1.65i)12-s + (−1.74 − 1.74i)13-s + (0.832 − 1.82i)14-s − 1.54·15-s + (−0.552 + 3.96i)16-s + 6.99·17-s + ⋯
L(s)  = 1  + (0.910 + 0.414i)2-s + (0.338 + 0.338i)3-s + (0.656 + 0.754i)4-s + (−0.590 + 0.590i)5-s + (0.167 + 0.448i)6-s − 0.537i·7-s + (0.284 + 0.958i)8-s − 0.770i·9-s + (−0.782 + 0.292i)10-s + (−0.213 + 0.213i)11-s + (−0.0331 + 0.477i)12-s + (−0.482 − 0.482i)13-s + (0.222 − 0.488i)14-s − 0.400·15-s + (−0.138 + 0.990i)16-s + 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.506 - 0.862i$
Analytic conductor: \(1.40536\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :1/2),\ 0.506 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64410 + 0.940891i\)
\(L(\frac12)\) \(\approx\) \(1.64410 + 0.940891i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.586i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-0.586 - 0.586i)T + 3iT^{2} \)
5 \( 1 + (1.32 - 1.32i)T - 5iT^{2} \)
7 \( 1 + 1.42iT - 7T^{2} \)
13 \( 1 + (1.74 + 1.74i)T + 13iT^{2} \)
17 \( 1 - 6.99T + 17T^{2} \)
19 \( 1 + (3.48 + 3.48i)T + 19iT^{2} \)
23 \( 1 + 4.79iT - 23T^{2} \)
29 \( 1 + (-3.38 - 3.38i)T + 29iT^{2} \)
31 \( 1 + 8.04T + 31T^{2} \)
37 \( 1 + (4.09 - 4.09i)T - 37iT^{2} \)
41 \( 1 + 5.85iT - 41T^{2} \)
43 \( 1 + (3.09 - 3.09i)T - 43iT^{2} \)
47 \( 1 + 8.12T + 47T^{2} \)
53 \( 1 + (-0.759 + 0.759i)T - 53iT^{2} \)
59 \( 1 + (0.513 - 0.513i)T - 59iT^{2} \)
61 \( 1 + (-0.211 - 0.211i)T + 61iT^{2} \)
67 \( 1 + (-0.116 - 0.116i)T + 67iT^{2} \)
71 \( 1 - 9.46iT - 71T^{2} \)
73 \( 1 - 7.55iT - 73T^{2} \)
79 \( 1 - 8.55T + 79T^{2} \)
83 \( 1 + (-7.36 - 7.36i)T + 83iT^{2} \)
89 \( 1 + 2.68iT - 89T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−12.80747907346369996469753564601, −12.15534748108069616514181036369, −10.98985508231822278945888708155, −10.07628587502780602349513345552, −8.543911677762268319955564594942, −7.44989553161865205178339994401, −6.67251551804783985519901297433, −5.17890179523264759313165165811, −3.87076134641849719957577952545, −2.99754421876959378699347313631, 1.90901546532702170384354139076, 3.46174778020870897351321714803, 4.84743254656332924167516206599, 5.82379556150558926564632219376, 7.40319265247117702815011598845, 8.280779018005384455581630284630, 9.691577621744629240216444275942, 10.78893188738063612379133266093, 11.99493681703418838648522486766, 12.40388000193432887267715346599

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