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Properties

Label 2-176-16.13-c1-0-9
Degree $2$
Conductor $176$
Sign $-0.0643 - 0.997i$
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  + (0.686 + 1.23i)2-s + (1.72 + 1.72i)3-s + (−1.05 + 1.69i)4-s + (0.893 − 0.893i)5-s + (−0.949 + 3.32i)6-s − 2.98i·7-s + (−2.82 − 0.141i)8-s + 2.97i·9-s + (1.71 + 0.491i)10-s + (0.707 − 0.707i)11-s + (−4.76 + 1.10i)12-s + (−3.01 − 3.01i)13-s + (3.69 − 2.05i)14-s + 3.08·15-s + (−1.76 − 3.58i)16-s − 6.89·17-s + ⋯
L(s)  = 1  + (0.485 + 0.874i)2-s + (0.997 + 0.997i)3-s + (−0.528 + 0.848i)4-s + (0.399 − 0.399i)5-s + (−0.387 + 1.35i)6-s − 1.12i·7-s + (−0.998 − 0.0498i)8-s + 0.990i·9-s + (0.543 + 0.155i)10-s + (0.213 − 0.213i)11-s + (−1.37 + 0.319i)12-s + (−0.836 − 0.836i)13-s + (0.987 − 0.548i)14-s + 0.797·15-s + (−0.441 − 0.897i)16-s − 1.67·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0643 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0643 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $-0.0643 - 0.997i$
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: \((2,\ 176,\ (\ :1/2),\ -0.0643 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24582 + 1.32871i\)
\(L(\frac12)\) \(\approx\) \(1.24582 + 1.32871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.686 - 1.23i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-1.72 - 1.72i)T + 3iT^{2} \)
5 \( 1 + (-0.893 + 0.893i)T - 5iT^{2} \)
7 \( 1 + 2.98iT - 7T^{2} \)
13 \( 1 + (3.01 + 3.01i)T + 13iT^{2} \)
17 \( 1 + 6.89T + 17T^{2} \)
19 \( 1 + (-2.03 - 2.03i)T + 19iT^{2} \)
23 \( 1 - 1.49iT - 23T^{2} \)
29 \( 1 + (-4.14 - 4.14i)T + 29iT^{2} \)
31 \( 1 - 1.31T + 31T^{2} \)
37 \( 1 + (5.93 - 5.93i)T - 37iT^{2} \)
41 \( 1 - 7.06iT - 41T^{2} \)
43 \( 1 + (-4.04 + 4.04i)T - 43iT^{2} \)
47 \( 1 - 5.17T + 47T^{2} \)
53 \( 1 + (-5.66 + 5.66i)T - 53iT^{2} \)
59 \( 1 + (3.49 - 3.49i)T - 59iT^{2} \)
61 \( 1 + (9.54 + 9.54i)T + 61iT^{2} \)
67 \( 1 + (9.71 + 9.71i)T + 67iT^{2} \)
71 \( 1 - 6.31iT - 71T^{2} \)
73 \( 1 + 0.614iT - 73T^{2} \)
79 \( 1 - 4.27T + 79T^{2} \)
83 \( 1 + (-10.0 - 10.0i)T + 83iT^{2} \)
89 \( 1 - 8.30iT - 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−13.58075706313120790338486741151, −12.32590413216656043851055322111, −10.70179471617157393551268365876, −9.680996325710419557671846249666, −8.871098725815509449181883181410, −7.890417972069713928131887313920, −6.76282714872086456593680194083, −5.14344183307855974043668680200, −4.23506793504067099525347261647, −3.11467363920941982765449226178, 2.16573423032044620567132649621, 2.59724017618102480365786837999, 4.51539129929294874218604040203, 6.10405427086635877474949698739, 7.13097221524921946569901627972, 8.802100707488329497289442385652, 9.187651422618060523321095278239, 10.56049547781826065551650926169, 11.88505520770658338534254085905, 12.38590735691898438273642780639

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