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Properties

Label 2-176-16.13-c1-0-11
Degree $2$
Conductor $176$
Sign $0.782 + 0.622i$
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  + (−0.0986 + 1.41i)2-s + (−1.15 − 1.15i)3-s + (−1.98 − 0.278i)4-s + (0.731 − 0.731i)5-s + (1.74 − 1.51i)6-s − 2.61i·7-s + (0.587 − 2.76i)8-s − 0.330i·9-s + (0.960 + 1.10i)10-s + (0.707 − 0.707i)11-s + (1.96 + 2.60i)12-s + (0.731 + 0.731i)13-s + (3.68 + 0.257i)14-s − 1.69·15-s + (3.84 + 1.10i)16-s + 3.67·17-s + ⋯
L(s)  = 1  + (−0.0697 + 0.997i)2-s + (−0.666 − 0.666i)3-s + (−0.990 − 0.139i)4-s + (0.327 − 0.327i)5-s + (0.711 − 0.618i)6-s − 0.987i·7-s + (0.207 − 0.978i)8-s − 0.110i·9-s + (0.303 + 0.349i)10-s + (0.213 − 0.213i)11-s + (0.567 + 0.753i)12-s + (0.202 + 0.202i)13-s + (0.984 + 0.0688i)14-s − 0.436·15-s + (0.961 + 0.275i)16-s + 0.891·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.782 + 0.622i$
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: \((2,\ 176,\ (\ :1/2),\ 0.782 + 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.763710 - 0.266658i\)
\(L(\frac12)\) \(\approx\) \(0.763710 - 0.266658i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0986 - 1.41i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (1.15 + 1.15i)T + 3iT^{2} \)
5 \( 1 + (-0.731 + 0.731i)T - 5iT^{2} \)
7 \( 1 + 2.61iT - 7T^{2} \)
13 \( 1 + (-0.731 - 0.731i)T + 13iT^{2} \)
17 \( 1 - 3.67T + 17T^{2} \)
19 \( 1 + (4.77 + 4.77i)T + 19iT^{2} \)
23 \( 1 + 2.84iT - 23T^{2} \)
29 \( 1 + (2.73 + 2.73i)T + 29iT^{2} \)
31 \( 1 - 5.05T + 31T^{2} \)
37 \( 1 + (2.69 - 2.69i)T - 37iT^{2} \)
41 \( 1 + 0.925iT - 41T^{2} \)
43 \( 1 + (6.27 - 6.27i)T - 43iT^{2} \)
47 \( 1 - 8.46T + 47T^{2} \)
53 \( 1 + (-7.66 + 7.66i)T - 53iT^{2} \)
59 \( 1 + (0.767 - 0.767i)T - 59iT^{2} \)
61 \( 1 + (-2.50 - 2.50i)T + 61iT^{2} \)
67 \( 1 + (-10.0 - 10.0i)T + 67iT^{2} \)
71 \( 1 + 3.26iT - 71T^{2} \)
73 \( 1 - 9.75iT - 73T^{2} \)
79 \( 1 - 9.41T + 79T^{2} \)
83 \( 1 + (-3.98 - 3.98i)T + 83iT^{2} \)
89 \( 1 - 3.97iT - 89T^{2} \)
97 \( 1 - 8.43T + 97T^{2} \)
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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

−12.95641356101840444234522058602, −11.76034628410419309880676163441, −10.52134182855441478016030397524, −9.420056780277730839477120391134, −8.313633997645623071888906575375, −7.09552630969398261198850049219, −6.45987677135522013919464044420, −5.33326167249626516629319184606, −3.99773632251735941035473924256, −0.894108257152490079491108928973, 2.17642588971680129486025370049, 3.79000449064159913077847215831, 5.16090906827458852857779240536, 6.00371236695809184936191408403, 8.015383968344440752285621607948, 9.113502007361465798562445679495, 10.20122960426621778167607050085, 10.64083119606289027637576800557, 11.86480603868010848166972073891, 12.36455524157651302807144097301

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