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Properties

Label 2-896-16.5-c1-0-14
Degree $2$
Conductor $896$
Sign $0.725 + 0.687i$
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  + (0.715 − 0.715i)3-s + (−0.867 − 0.867i)5-s − i·7-s + 1.97i·9-s + (2.97 + 2.97i)11-s + (2.02 − 2.02i)13-s − 1.24·15-s + 0.264·17-s + (4.53 − 4.53i)19-s + (−0.715 − 0.715i)21-s + 1.54i·23-s − 3.49i·25-s + (3.55 + 3.55i)27-s + (0.328 − 0.328i)29-s + 6.04·31-s + ⋯
L(s)  = 1  + (0.412 − 0.412i)3-s + (−0.388 − 0.388i)5-s − 0.377i·7-s + 0.658i·9-s + (0.897 + 0.897i)11-s + (0.560 − 0.560i)13-s − 0.320·15-s + 0.0641·17-s + (1.04 − 1.04i)19-s + (−0.156 − 0.156i)21-s + 0.322i·23-s − 0.698i·25-s + (0.685 + 0.685i)27-s + (0.0609 − 0.0609i)29-s + 1.08·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.725 + 0.687i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.725 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70102 - 0.677868i\)
\(L(\frac12)\) \(\approx\) \(1.70102 - 0.677868i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 + (-0.715 + 0.715i)T - 3iT^{2} \)
5 \( 1 + (0.867 + 0.867i)T + 5iT^{2} \)
11 \( 1 + (-2.97 - 2.97i)T + 11iT^{2} \)
13 \( 1 + (-2.02 + 2.02i)T - 13iT^{2} \)
17 \( 1 - 0.264T + 17T^{2} \)
19 \( 1 + (-4.53 + 4.53i)T - 19iT^{2} \)
23 \( 1 - 1.54iT - 23T^{2} \)
29 \( 1 + (-0.328 + 0.328i)T - 29iT^{2} \)
31 \( 1 - 6.04T + 31T^{2} \)
37 \( 1 + (6.64 + 6.64i)T + 37iT^{2} \)
41 \( 1 + 11.0iT - 41T^{2} \)
43 \( 1 + (3.38 + 3.38i)T + 43iT^{2} \)
47 \( 1 - 3.12T + 47T^{2} \)
53 \( 1 + (0.430 + 0.430i)T + 53iT^{2} \)
59 \( 1 + (-4.62 - 4.62i)T + 59iT^{2} \)
61 \( 1 + (4.86 - 4.86i)T - 61iT^{2} \)
67 \( 1 + (3.34 - 3.34i)T - 67iT^{2} \)
71 \( 1 - 9.03iT - 71T^{2} \)
73 \( 1 - 14.8iT - 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 + (-0.715 + 0.715i)T - 83iT^{2} \)
89 \( 1 + 10.9iT - 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−10.02391705521990138165881160375, −8.990764082238309494734659582766, −8.368123522895199938777483274600, −7.36983045569308758076261905271, −6.95460919543145714749946752645, −5.56066487937968876860302813168, −4.60851046440073446247859194683, −3.65589266559999535045640963371, −2.35487363157129045498342978195, −1.03498732219370290647029361941, 1.32867866622646075407258444602, 3.20493705843735616770708806843, 3.55533386793003101077169099469, 4.75877843745015782060442550034, 6.12228593858123645335624202747, 6.58860814767637894987791366953, 7.88135977682997493853623608593, 8.620840299760090629693307058616, 9.350551987655423578822378523189, 10.04464223007319151021540640159

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