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Properties

Label 2-176-44.43-c1-0-1
Degree $2$
Conductor $176$
Sign $-i$
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  + 3.31i·3-s + 3·5-s − 8·9-s − 3.31i·11-s + 9.94i·15-s + 3.31i·23-s + 4·25-s − 16.5i·27-s − 9.94i·31-s + 11·33-s + 7·37-s − 24·45-s + 6.63i·47-s − 7·49-s + 6·53-s + ⋯
L(s)  = 1  + 1.91i·3-s + 1.34·5-s − 2.66·9-s − 1.00i·11-s + 2.56i·15-s + 0.691i·23-s + 0.800·25-s − 3.19i·27-s − 1.78i·31-s + 1.91·33-s + 1.15·37-s − 3.57·45-s + 0.967i·47-s − 49-s + 0.824·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $-i$
Analytic conductor: \(1.40536\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.932324 + 0.932324i\)
\(L(\frac12)\) \(\approx\) \(0.932324 + 0.932324i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + 3.31iT \)
good3 \( 1 - 3.31iT - 3T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 3.31iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 9.94iT - 31T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 6.63iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 3.31iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 9.94iT - 67T^{2} \)
71 \( 1 - 16.5iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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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

−13.26689574460251639152912224510, −11.49198379525622179553040582408, −10.81544137256835978847799455611, −9.748489922897669210009610733974, −9.412776763383933013265662140945, −8.265812388964835060097532459611, −6.04770729300202012507919444525, −5.46736213803717728599297204420, −4.14541389254925606064779588488, −2.79614747922257642985836490734, 1.54947310926681241241979002733, 2.57926410663713843923971601300, 5.26289601900205623389920953260, 6.36473958059617845155195121868, 7.02685845751682152482628040032, 8.200933396978128641688989042432, 9.295598080071927005493129245652, 10.51289893843233578366461806575, 11.85656608214118645293502039957, 12.66488246036728006839149801352

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