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Properties

Label 2-176-11.3-c1-0-0
Degree $2$
Conductor $176$
Sign $0.205 - 0.978i$
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  + (0.632 + 1.94i)3-s + (0.132 + 0.0962i)5-s + (−1.08 + 3.32i)7-s + (−0.962 + 0.699i)9-s + (0.605 − 3.26i)11-s + (−2.17 + 1.58i)13-s + (−0.103 + 0.318i)15-s + (6.12 + 4.45i)17-s + (−1.42 − 4.39i)19-s − 7.16·21-s + 0.706·23-s + (−1.53 − 4.72i)25-s + (2.99 + 2.17i)27-s + (−0.317 + 0.978i)29-s + (4.36 − 3.17i)31-s + ⋯
L(s)  = 1  + (0.365 + 1.12i)3-s + (0.0592 + 0.0430i)5-s + (−0.408 + 1.25i)7-s + (−0.320 + 0.233i)9-s + (0.182 − 0.983i)11-s + (−0.604 + 0.439i)13-s + (−0.0267 + 0.0823i)15-s + (1.48 + 1.08i)17-s + (−0.327 − 1.00i)19-s − 1.56·21-s + 0.147·23-s + (−0.307 − 0.945i)25-s + (0.577 + 0.419i)27-s + (−0.0590 + 0.181i)29-s + (0.784 − 0.570i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.205 - 0.978i$
Analytic conductor: \(1.40536\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :1/2),\ 0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.978595 + 0.794186i\)
\(L(\frac12)\) \(\approx\) \(0.978595 + 0.794186i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.605 + 3.26i)T \)
good3 \( 1 + (-0.632 - 1.94i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.132 - 0.0962i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.08 - 3.32i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.17 - 1.58i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-6.12 - 4.45i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.42 + 4.39i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.706T + 23T^{2} \)
29 \( 1 + (0.317 - 0.978i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.36 + 3.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.58 + 11.0i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.0867 + 0.267i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.31T + 43T^{2} \)
47 \( 1 + (1.80 + 5.54i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.98 - 5.80i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.954 + 2.93i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.60 + 4.07i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 1.91T + 67T^{2} \)
71 \( 1 + (-9.84 - 7.15i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.51 - 10.8i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.39 - 1.01i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.81 - 2.76i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.31T + 89T^{2} \)
97 \( 1 + (5.66 - 4.11i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.78910917917367609455976449908, −11.93783356083992030203952430430, −10.78545077875222608301362614272, −9.755851615318848197798348598797, −9.079029026967876296918724657305, −8.152225243586906976660095379235, −6.37491048399335312868704848626, −5.33896613363675125677552180647, −3.93642021275947198122313611779, −2.71849584237390381850445638510, 1.35581194759338434609970806089, 3.16636848561437883201834090916, 4.80020257559731952294416070601, 6.49830332980487890222439246367, 7.44702989004712179308848779685, 7.903697349301780603942310972774, 9.703539046774580095873210925889, 10.24371291211192062639792896442, 11.85734630982709314041070103355, 12.58892775332025253809914477572

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