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Properties

Label 2-896-56.19-c1-0-4
Degree $2$
Conductor $896$
Sign $-0.293 - 0.955i$
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  + (−2.63 + 1.52i)3-s + (0.866 − 1.5i)5-s + (−2.14 − 1.54i)7-s + (3.12 − 5.40i)9-s + (1.09 + 1.88i)11-s − 5.91·13-s + 5.26i·15-s + (3.62 − 2.09i)17-s + (3.27 + 1.88i)19-s + (8.00 + 0.792i)21-s + (1.88 + 1.09i)23-s + (1 + 1.73i)25-s + 9.85i·27-s + 7.41i·29-s + (−4.03 − 6.99i)31-s + ⋯
L(s)  = 1  + (−1.52 + 0.877i)3-s + (0.387 − 0.670i)5-s + (−0.812 − 0.582i)7-s + (1.04 − 1.80i)9-s + (0.328 + 0.569i)11-s − 1.64·13-s + 1.35i·15-s + (0.878 − 0.507i)17-s + (0.750 + 0.433i)19-s + (1.74 + 0.173i)21-s + (0.393 + 0.227i)23-s + (0.200 + 0.346i)25-s + 1.89i·27-s + 1.37i·29-s + (−0.725 − 1.25i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.293 - 0.955i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (831, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -0.293 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.322436 + 0.436389i\)
\(L(\frac12)\) \(\approx\) \(0.322436 + 0.436389i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 + 1.54i)T \)
good3 \( 1 + (2.63 - 1.52i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.09 - 1.88i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.91T + 13T^{2} \)
17 \( 1 + (-3.62 + 2.09i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.27 - 1.88i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.88 - 1.09i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.41iT - 29T^{2} \)
31 \( 1 + (4.03 + 6.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.07 - 0.621i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.34iT - 41T^{2} \)
43 \( 1 + 6.16T + 43T^{2} \)
47 \( 1 + (4.56 - 7.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.37 + 0.792i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.63 - 1.52i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.07 - 1.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.81 - 8.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.72iT - 71T^{2} \)
73 \( 1 + (11.7 - 6.77i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.23 - 4.17i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.1iT - 83T^{2} \)
89 \( 1 + (-9.98 - 5.76i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.91iT - 97T^{2} \)
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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

−10.07938397899532797849869924456, −9.774187445575017576063351069560, −9.271060791417888762057655120095, −7.50111387435833763728464923591, −6.86697169892012126635842164083, −5.75735537316577252605866864595, −5.10970360156660394353017419392, −4.44625708117886304559211038032, −3.24248162793664234062909179608, −1.12425089420593870193687227625, 0.36611233181018548654352250629, 2.04732460998470963388898099145, 3.21617369995409276465379722985, 4.99082349993636951247216649932, 5.63616180090014801676431322694, 6.47265604777575826287426902590, 6.95859298527713492047022523002, 7.84631543214896931650704265223, 9.231585476837999387232595867713, 10.13370931365055955569205323588

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