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Properties

Label 2-896-112.109-c1-0-23
Degree $2$
Conductor $896$
Sign $0.889 + 0.456i$
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  + (2.91 − 0.781i)3-s + (0.745 + 0.199i)5-s + (2.51 − 0.830i)7-s + (5.28 − 3.05i)9-s + (−0.333 − 1.24i)11-s + (0.919 + 0.919i)13-s + 2.32·15-s + (−3.95 + 6.85i)17-s + (0.478 − 1.78i)19-s + (6.67 − 4.38i)21-s + (−3.33 + 1.92i)23-s + (−3.81 − 2.20i)25-s + (6.63 − 6.63i)27-s + (−5.25 − 5.25i)29-s + (−2.44 + 4.23i)31-s + ⋯
L(s)  = 1  + (1.68 − 0.450i)3-s + (0.333 + 0.0893i)5-s + (0.949 − 0.313i)7-s + (1.76 − 1.01i)9-s + (−0.100 − 0.374i)11-s + (0.254 + 0.254i)13-s + 0.601·15-s + (−0.959 + 1.66i)17-s + (0.109 − 0.409i)19-s + (1.45 − 0.956i)21-s + (−0.694 + 0.401i)23-s + (−0.762 − 0.440i)25-s + (1.27 − 1.27i)27-s + (−0.975 − 0.975i)29-s + (−0.439 + 0.761i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.889 + 0.456i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.889 + 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.03427 - 0.732428i\)
\(L(\frac12)\) \(\approx\) \(3.03427 - 0.732428i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.51 + 0.830i)T \)
good3 \( 1 + (-2.91 + 0.781i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.745 - 0.199i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.333 + 1.24i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.919 - 0.919i)T + 13iT^{2} \)
17 \( 1 + (3.95 - 6.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.478 + 1.78i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.33 - 1.92i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.25 + 5.25i)T + 29iT^{2} \)
31 \( 1 + (2.44 - 4.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.28 - 0.343i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.84iT - 41T^{2} \)
43 \( 1 + (-0.585 + 0.585i)T - 43iT^{2} \)
47 \( 1 + (-2.86 - 4.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.54 + 9.51i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.39 - 8.93i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.71 + 6.38i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-5.94 + 1.59i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.99iT - 71T^{2} \)
73 \( 1 + (6.69 + 3.86i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.63 - 8.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.78 + 4.78i)T + 83iT^{2} \)
89 \( 1 + (1.84 - 1.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.01T + 97T^{2} \)
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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

−9.855531779683679155191110467543, −9.006386429647748110948811518320, −8.272970362839509547372000580545, −7.85101359150701670283570928910, −6.84664729127308745735745374960, −5.84427600560926007041687541185, −4.31460356709420207688898051621, −3.64680571009570608316938062994, −2.27195428461779461663862730462, −1.62713997769893765840448341625, 1.84999637659877433000244447293, 2.56237385318492432666698216985, 3.77807970583799908888412074432, 4.65969334636958412272390405023, 5.61262439383639315203213781495, 7.19800050229829702969737737026, 7.76769056254917168706572306823, 8.655771918395801722828964862055, 9.229337755314025904229347041889, 9.861398725765448874528850645858

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