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Properties

Label 2-896-112.53-c1-0-5
Degree $2$
Conductor $896$
Sign $-0.658 - 0.752i$
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  + (−0.224 − 0.839i)3-s + (−0.847 + 3.16i)5-s + (−0.654 + 2.56i)7-s + (1.94 − 1.12i)9-s + (−2.87 + 0.769i)11-s + (3.63 − 3.63i)13-s + 2.84·15-s + (−1.81 + 3.14i)17-s + (−1.60 − 0.429i)19-s + (2.29 − 0.0267i)21-s + (−5.33 + 3.08i)23-s + (−4.95 − 2.86i)25-s + (−3.22 − 3.22i)27-s + (−5.10 + 5.10i)29-s + (−1.00 + 1.74i)31-s + ⋯
L(s)  = 1  + (−0.129 − 0.484i)3-s + (−0.379 + 1.41i)5-s + (−0.247 + 0.968i)7-s + (0.648 − 0.374i)9-s + (−0.865 + 0.231i)11-s + (1.00 − 1.00i)13-s + 0.734·15-s + (−0.441 + 0.763i)17-s + (−0.367 − 0.0985i)19-s + (0.501 − 0.00584i)21-s + (−1.11 + 0.642i)23-s + (−0.991 − 0.572i)25-s + (−0.620 − 0.620i)27-s + (−0.947 + 0.947i)29-s + (−0.180 + 0.313i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.658 - 0.752i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.343466 + 0.757391i\)
\(L(\frac12)\) \(\approx\) \(0.343466 + 0.757391i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.654 - 2.56i)T \)
good3 \( 1 + (0.224 + 0.839i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.847 - 3.16i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.87 - 0.769i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.63 + 3.63i)T - 13iT^{2} \)
17 \( 1 + (1.81 - 3.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.60 + 0.429i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.33 - 3.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.10 - 5.10i)T - 29iT^{2} \)
31 \( 1 + (1.00 - 1.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.49 - 5.57i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.71iT - 41T^{2} \)
43 \( 1 + (-2.91 - 2.91i)T + 43iT^{2} \)
47 \( 1 + (5.06 + 8.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.68 - 0.986i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.64 + 0.977i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.54 - 1.75i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.57 - 5.88i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.55iT - 71T^{2} \)
73 \( 1 + (0.989 + 0.571i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.120 - 0.209i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.459 + 0.459i)T - 83iT^{2} \)
89 \( 1 + (-3.76 + 2.17i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.80T + 97T^{2} \)
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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

−10.45412758866137876543007145896, −9.814270466809949747705471854108, −8.525603297854488384981832191047, −7.84421268441670487106777494539, −6.92300304139422143991027562555, −6.25220540948350887864717875317, −5.45364029890972939051900815541, −3.83962122562928807490094439817, −3.03609445051295484791906966925, −1.87986248303447746216305285018, 0.39182357569956687501937451495, 1.89810589701856496351631387059, 3.89145870497201008153252023889, 4.29667647823484869719753620964, 5.15899178931277967471017750330, 6.28616113370343178318596873154, 7.45745077645025919538860550390, 8.123615095968224450120569645821, 9.076248695589599173521186667824, 9.723683748791330940947630055157

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