Properties

Label 2-896-16.13-c1-0-13
Degree $2$
Conductor $896$
Sign $0.994 + 0.103i$
Analytic cond. $7.15459$
Root an. cond. $2.67480$
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  + (0.631 + 0.631i)3-s + (2.34 − 2.34i)5-s + i·7-s − 2.20i·9-s + (−2.18 + 2.18i)11-s + (4.03 + 4.03i)13-s + 2.95·15-s + 0.347·17-s + (4.26 + 4.26i)19-s + (−0.631 + 0.631i)21-s − 6.23i·23-s − 5.97i·25-s + (3.28 − 3.28i)27-s + (−1.21 − 1.21i)29-s + 1.26·31-s + ⋯
L(s)  = 1  + (0.364 + 0.364i)3-s + (1.04 − 1.04i)5-s + 0.377i·7-s − 0.734i·9-s + (−0.658 + 0.658i)11-s + (1.11 + 1.11i)13-s + 0.763·15-s + 0.0843·17-s + (0.978 + 0.978i)19-s + (−0.137 + 0.137i)21-s − 1.30i·23-s − 1.19i·25-s + (0.632 − 0.632i)27-s + (−0.226 − 0.226i)29-s + 0.226·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.994 + 0.103i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.994 + 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19678 - 0.114382i\)
\(L(\frac12)\) \(\approx\) \(2.19678 - 0.114382i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - iT \)
good3 \( 1 + (-0.631 - 0.631i)T + 3iT^{2} \)
5 \( 1 + (-2.34 + 2.34i)T - 5iT^{2} \)
11 \( 1 + (2.18 - 2.18i)T - 11iT^{2} \)
13 \( 1 + (-4.03 - 4.03i)T + 13iT^{2} \)
17 \( 1 - 0.347T + 17T^{2} \)
19 \( 1 + (-4.26 - 4.26i)T + 19iT^{2} \)
23 \( 1 + 6.23iT - 23T^{2} \)
29 \( 1 + (1.21 + 1.21i)T + 29iT^{2} \)
31 \( 1 - 1.26T + 31T^{2} \)
37 \( 1 + (-6.42 + 6.42i)T - 37iT^{2} \)
41 \( 1 + 2.68iT - 41T^{2} \)
43 \( 1 + (-4.05 + 4.05i)T - 43iT^{2} \)
47 \( 1 + 4.64T + 47T^{2} \)
53 \( 1 + (8.44 - 8.44i)T - 53iT^{2} \)
59 \( 1 + (5.17 - 5.17i)T - 59iT^{2} \)
61 \( 1 + (-0.00533 - 0.00533i)T + 61iT^{2} \)
67 \( 1 + (-3.02 - 3.02i)T + 67iT^{2} \)
71 \( 1 + 0.828iT - 71T^{2} \)
73 \( 1 + 6.25iT - 73T^{2} \)
79 \( 1 - 0.755T + 79T^{2} \)
83 \( 1 + (3.66 + 3.66i)T + 83iT^{2} \)
89 \( 1 - 6.24iT - 89T^{2} \)
97 \( 1 - 2.18T + 97T^{2} \)
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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

−9.822211280903922276650103688904, −9.232527058992209017322270251216, −8.766552472614115056744451393797, −7.75550289073804827179279823085, −6.39373518570273113647032467354, −5.78765709674525399144604809817, −4.75053569800118212397672654240, −3.86929741722000825466258722866, −2.44368621687686454390564663175, −1.29185061519324759713733918141, 1.37812326826724319342326335904, 2.78692763475712642515944108396, 3.26396595290579944582701522718, 5.08006530027914274236332980091, 5.83652958541924072837546832543, 6.70054441590683815882349867372, 7.69499966946087164985548643513, 8.203762543386261352768276361077, 9.471356192864126975440512694454, 10.14031829135577518097525054407

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