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Properties

Label 2-896-112.3-c1-0-16
Degree $2$
Conductor $896$
Sign $0.974 - 0.224i$
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  + (3.08 − 0.825i)3-s + (−0.501 + 1.86i)5-s + (2.17 + 1.49i)7-s + (6.21 − 3.58i)9-s + (−3.52 + 0.944i)11-s + (0.372 − 0.372i)13-s + 6.17i·15-s + (2.39 + 1.38i)17-s + (−0.431 + 1.60i)19-s + (7.95 + 2.82i)21-s + (−1.27 − 2.20i)23-s + (1.08 + 0.626i)25-s + (9.41 − 9.41i)27-s + (−2.14 − 2.14i)29-s + (2.74 − 4.75i)31-s + ⋯
L(s)  = 1  + (1.77 − 0.476i)3-s + (−0.224 + 0.836i)5-s + (0.823 + 0.566i)7-s + (2.07 − 1.19i)9-s + (−1.06 + 0.284i)11-s + (0.103 − 0.103i)13-s + 1.59i·15-s + (0.582 + 0.336i)17-s + (−0.0989 + 0.369i)19-s + (1.73 + 0.615i)21-s + (−0.265 − 0.459i)23-s + (0.216 + 0.125i)25-s + (1.81 − 1.81i)27-s + (−0.397 − 0.397i)29-s + (0.493 − 0.854i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.974 - 0.224i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.974 - 0.224i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.91739 + 0.331931i\)
\(L(\frac12)\) \(\approx\) \(2.91739 + 0.331931i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 - 1.49i)T \)
good3 \( 1 + (-3.08 + 0.825i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (0.501 - 1.86i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.52 - 0.944i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.372 + 0.372i)T - 13iT^{2} \)
17 \( 1 + (-2.39 - 1.38i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.431 - 1.60i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.27 + 2.20i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.14 + 2.14i)T + 29iT^{2} \)
31 \( 1 + (-2.74 + 4.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.53 + 1.21i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 3.95T + 41T^{2} \)
43 \( 1 + (-4.29 - 4.29i)T + 43iT^{2} \)
47 \( 1 + (1.85 + 3.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.74 - 6.49i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.20 + 8.21i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.378 + 0.101i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (3.72 + 13.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.98T + 71T^{2} \)
73 \( 1 + (0.766 - 1.32i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.47 - 2.00i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.67 + 3.67i)T + 83iT^{2} \)
89 \( 1 + (3.35 + 5.80i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.52iT - 97T^{2} \)
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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

−10.06042947672849923802699792132, −9.125408208946168584747089688531, −8.220805055023746209795272616600, −7.83837040152304229341859007555, −7.13466703497099034923840055838, −5.94574974375334986433214504502, −4.57808450800711032297337530538, −3.41875967152918348656880167748, −2.62553724589555194660004543098, −1.78804373236839442207642130433, 1.40198312535924294664077436701, 2.67956121099178028120551865375, 3.67325461970658863098271734516, 4.61041378274306778410755117289, 5.27356124227991284901760654629, 7.19478418262731675204019677772, 7.78323263177421507045043506470, 8.560619810598034235727747378060, 8.898980712123431740248619025095, 10.07001019268713748975714627275

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