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Properties

Label 2-176-16.13-c1-0-6
Degree $2$
Conductor $176$
Sign $0.962 - 0.271i$
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  + (−1.15 − 0.819i)2-s + (2.04 + 2.04i)3-s + (0.655 + 1.88i)4-s + (0.814 − 0.814i)5-s + (−0.680 − 4.03i)6-s − 2.42i·7-s + (0.794 − 2.71i)8-s + 5.38i·9-s + (−1.60 + 0.270i)10-s + (−0.707 + 0.707i)11-s + (−2.52 + 5.21i)12-s + (4.19 + 4.19i)13-s + (−1.98 + 2.79i)14-s + 3.33·15-s + (−3.14 + 2.47i)16-s − 3.70·17-s + ⋯
L(s)  = 1  + (−0.814 − 0.579i)2-s + (1.18 + 1.18i)3-s + (0.327 + 0.944i)4-s + (0.364 − 0.364i)5-s + (−0.277 − 1.64i)6-s − 0.916i·7-s + (0.280 − 0.959i)8-s + 1.79i·9-s + (−0.508 + 0.0856i)10-s + (−0.213 + 0.213i)11-s + (−0.729 + 1.50i)12-s + (1.16 + 1.16i)13-s + (−0.531 + 0.746i)14-s + 0.861·15-s + (−0.785 + 0.619i)16-s − 0.897·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.962 - 0.271i$
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: \((2,\ 176,\ (\ :1/2),\ 0.962 - 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17683 + 0.162845i\)
\(L(\frac12)\) \(\approx\) \(1.17683 + 0.162845i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 + 0.819i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-2.04 - 2.04i)T + 3iT^{2} \)
5 \( 1 + (-0.814 + 0.814i)T - 5iT^{2} \)
7 \( 1 + 2.42iT - 7T^{2} \)
13 \( 1 + (-4.19 - 4.19i)T + 13iT^{2} \)
17 \( 1 + 3.70T + 17T^{2} \)
19 \( 1 + (1.91 + 1.91i)T + 19iT^{2} \)
23 \( 1 + 5.56iT - 23T^{2} \)
29 \( 1 + (2.07 + 2.07i)T + 29iT^{2} \)
31 \( 1 + 5.32T + 31T^{2} \)
37 \( 1 + (-5.40 + 5.40i)T - 37iT^{2} \)
41 \( 1 + 7.68iT - 41T^{2} \)
43 \( 1 + (4.92 - 4.92i)T - 43iT^{2} \)
47 \( 1 + 7.29T + 47T^{2} \)
53 \( 1 + (1.72 - 1.72i)T - 53iT^{2} \)
59 \( 1 + (-8.02 + 8.02i)T - 59iT^{2} \)
61 \( 1 + (5.82 + 5.82i)T + 61iT^{2} \)
67 \( 1 + (5.59 + 5.59i)T + 67iT^{2} \)
71 \( 1 - 1.42iT - 71T^{2} \)
73 \( 1 - 16.1iT - 73T^{2} \)
79 \( 1 - 5.56T + 79T^{2} \)
83 \( 1 + (1.72 + 1.72i)T + 83iT^{2} \)
89 \( 1 - 13.5iT - 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−12.97676075593630706017791200359, −11.16758543284843849600116880914, −10.65908911281381507404730393493, −9.514474377678400264101407590301, −9.021154439873771601780343785782, −8.148621503513454181717597114788, −6.80877787047934647394704447707, −4.45757414720440223529544503803, −3.70808041666335096041899082437, −2.10101360936661096144344541944, 1.72697433902593104414351060292, 2.97702609820403254570224821552, 5.74475702848951376418747157395, 6.54903079200240760987344201677, 7.76733561480333427416789921137, 8.456894456264853182987840103426, 9.170230490297137136463662494439, 10.43809476510334605405314854329, 11.64415358972326828448762195537, 13.08191051128765430948272357601

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