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Properties

Label 2-176-16.13-c1-0-5
Degree $2$
Conductor $176$
Sign $-0.0349 - 0.999i$
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  + (−0.398 + 1.35i)2-s + (1.34 + 1.34i)3-s + (−1.68 − 1.08i)4-s + (2.40 − 2.40i)5-s + (−2.36 + 1.28i)6-s + 4.88i·7-s + (2.13 − 1.85i)8-s + 0.617i·9-s + (2.30 + 4.22i)10-s + (−0.707 + 0.707i)11-s + (−0.807 − 3.71i)12-s + (−0.532 − 0.532i)13-s + (−6.63 − 1.94i)14-s + 6.47·15-s + (1.65 + 3.63i)16-s − 1.53·17-s + ⋯
L(s)  = 1  + (−0.281 + 0.959i)2-s + (0.776 + 0.776i)3-s + (−0.841 − 0.540i)4-s + (1.07 − 1.07i)5-s + (−0.963 + 0.526i)6-s + 1.84i·7-s + (0.756 − 0.654i)8-s + 0.205i·9-s + (0.729 + 1.33i)10-s + (−0.213 + 0.213i)11-s + (−0.233 − 1.07i)12-s + (−0.147 − 0.147i)13-s + (−1.77 − 0.520i)14-s + 1.67·15-s + (0.414 + 0.909i)16-s − 0.373·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0349 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0349 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $-0.0349 - 0.999i$
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: \((2,\ 176,\ (\ :1/2),\ -0.0349 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.919858 + 0.952621i\)
\(L(\frac12)\) \(\approx\) \(0.919858 + 0.952621i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.398 - 1.35i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-1.34 - 1.34i)T + 3iT^{2} \)
5 \( 1 + (-2.40 + 2.40i)T - 5iT^{2} \)
7 \( 1 - 4.88iT - 7T^{2} \)
13 \( 1 + (0.532 + 0.532i)T + 13iT^{2} \)
17 \( 1 + 1.53T + 17T^{2} \)
19 \( 1 + (5.98 + 5.98i)T + 19iT^{2} \)
23 \( 1 + 0.294iT - 23T^{2} \)
29 \( 1 + (-4.94 - 4.94i)T + 29iT^{2} \)
31 \( 1 + 1.43T + 31T^{2} \)
37 \( 1 + (-1.66 + 1.66i)T - 37iT^{2} \)
41 \( 1 + 4.09iT - 41T^{2} \)
43 \( 1 + (-4.95 + 4.95i)T - 43iT^{2} \)
47 \( 1 + 0.856T + 47T^{2} \)
53 \( 1 + (0.0858 - 0.0858i)T - 53iT^{2} \)
59 \( 1 + (3.37 - 3.37i)T - 59iT^{2} \)
61 \( 1 + (-0.868 - 0.868i)T + 61iT^{2} \)
67 \( 1 + (1.38 + 1.38i)T + 67iT^{2} \)
71 \( 1 + 2.83iT - 71T^{2} \)
73 \( 1 + 5.62iT - 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + (-5.47 - 5.47i)T + 83iT^{2} \)
89 \( 1 - 8.20iT - 89T^{2} \)
97 \( 1 + 7.42T + 97T^{2} \)
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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

−13.07783591748184503525258789462, −12.38505385942796857546648904289, −10.46977952398762626633273147056, −9.237643419813948929159312070146, −9.058116466208724162135138758684, −8.403208314338543503907724261204, −6.46175812807642293108345050852, −5.41411373689734329245770553560, −4.61177255816955383664206727863, −2.37464266775010534649877072284, 1.68141546167109694375102553908, 2.86847012305084734055732552729, 4.23055718196265783377401417162, 6.40374336129251711077323124573, 7.46628916504472737009173182451, 8.309071600457257413082240094890, 9.840965818023970096327922527481, 10.40903109923337750943706668295, 11.12689672367193008487539233128, 12.79487829352711133228027958087

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