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Properties

Label 2-896-112.109-c1-0-17
Degree $2$
Conductor $896$
Sign $0.508 + 0.861i$
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  + (−0.831 + 0.222i)3-s + (2.02 + 0.543i)5-s + (−2.63 − 0.233i)7-s + (−1.95 + 1.12i)9-s + (−1.03 − 3.85i)11-s + (0.990 + 0.990i)13-s − 1.80·15-s + (3.07 − 5.33i)17-s + (1.01 − 3.79i)19-s + (2.24 − 0.393i)21-s + (5.91 − 3.41i)23-s + (−0.514 − 0.297i)25-s + (3.20 − 3.20i)27-s + (3.83 + 3.83i)29-s + (−2.05 + 3.55i)31-s + ⋯
L(s)  = 1  + (−0.480 + 0.128i)3-s + (0.906 + 0.242i)5-s + (−0.996 − 0.0881i)7-s + (−0.652 + 0.376i)9-s + (−0.311 − 1.16i)11-s + (0.274 + 0.274i)13-s − 0.466·15-s + (0.746 − 1.29i)17-s + (0.233 − 0.870i)19-s + (0.489 − 0.0858i)21-s + (1.23 − 0.712i)23-s + (−0.102 − 0.0594i)25-s + (0.616 − 0.616i)27-s + (0.712 + 0.712i)29-s + (−0.368 + 0.638i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.508 + 0.861i$
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: \((2,\ 896,\ (\ :1/2),\ 0.508 + 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.995535 - 0.568472i\)
\(L(\frac12)\) \(\approx\) \(0.995535 - 0.568472i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 + 0.233i)T \)
good3 \( 1 + (0.831 - 0.222i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-2.02 - 0.543i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.03 + 3.85i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.990 - 0.990i)T + 13iT^{2} \)
17 \( 1 + (-3.07 + 5.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.01 + 3.79i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-5.91 + 3.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.83 - 3.83i)T + 29iT^{2} \)
31 \( 1 + (2.05 - 3.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0740 + 0.0198i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.68iT - 41T^{2} \)
43 \( 1 + (0.713 - 0.713i)T - 43iT^{2} \)
47 \( 1 + (-1.95 - 3.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.89 + 7.06i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.851 - 3.17i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.37 + 8.84i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.49 + 0.401i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.86iT - 71T^{2} \)
73 \( 1 + (-8.95 - 5.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.33 + 5.77i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.2 + 10.2i)T + 83iT^{2} \)
89 \( 1 + (1.16 - 0.671i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 18.7T + 97T^{2} \)
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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

−10.05341366996867625117404604493, −9.178616231473534114980461336502, −8.542817011859067650720323312706, −7.15888706168767916914989735411, −6.44416381408731965002213205308, −5.60648128647663751895326509757, −4.98493798504033280324534115748, −3.26038351146020815526407740359, −2.64199936394066732487213648262, −0.61981706900594756336997095353, 1.35514533493797597158188556884, 2.75849002443592218335710202919, 3.87440946929478191305084505654, 5.33573486036866213191395295166, 5.87447106190011532965135312817, 6.58486368088996676911497999329, 7.66427574834065421339563578884, 8.696222918544608033920284348194, 9.736096095852675838949489706418, 9.955821181080772076759509891804

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